mm-3319 === Subject: Re: Bring an arbitrary nxn Boolean matrix to a goal matrix (developing fitness test - help please) > 5) No row or column exchanges are allowed The following sequence exchanges columns/rows A and B A <= AxB, B <= AxB, A <= AxB So they are. Toby === Subject: Re: Bring an arbitrary nxn Boolean matrix to a goal matrix (developing fitness test - help please) | > 5) No row or column exchanges are allowed | | The following sequence exchanges columns/rows A and B | A <= AxB, B <= AxB, A <= AxB These are clearly XOR operations, not row exchanges, so they are indeed legal. It's just that the requirement is for people out there to not swap rows/columns like in the rref algorithm. What you have written out should be a natural consequence of an algorithm which 'solves' the matrix. Do you have any suggestions about my first requirement? Eric === Subject: History, Mystery and Chemistry of E=mc2. History, Mystery and Chemistry of E=mc2. The following can be regarded as the most three significant equations in physics. E= mc2 F=ma F =GmM/r2 All the significant and critical aspects of E= mc2 are explained in form of Frequently Asked Questions at Link http://physicsajay.sulekha.com/blog/post/2006/11/galileo-not-einstein-is-inv entor-of-second-postulate.htm AJAY SHARMA === Subject: Re: History, Mystery and Chemistry of E=mc2. > History, Mystery and Chemistry of E=mc2. The following can be regarded as the most three significant equations > in physics. > E= mc2 > F=ma > F =GmM/r2 > All the significant and critical aspects of E= mc2 > are explained in form of Frequently Asked Questions at > Link http://physicsajay.sulekha.com/blog/post/2006/11/galileo-not-einstein-is-inv e ntor-of-second-postulate.htm AJAY SHARMA --------------------------- here is the complete link http://physicsajay.sulekha.com/blog/post/2006/11/galileo-not-einstein-is-inv entor-of-second-postulate.htm === Subject: Re: History, Mystery and Chemistry of E=mc2. > History, Mystery and Chemistry of E=mc2. The following can be regarded as the most three significant equations > in physics. > E= mc2 > F=ma > F =GmM/r2 > All the significant and critical aspects of E= mc2 > are explained in form of Frequently Asked Questions at > Link http://physicsajay.sulekha.com/blog/post/2006/11/galileo-not-einstein-is-inv e ntor-of-second-postulate.htm AJAY SHARMA > --------------------------- > here is the complete link http://physicsajay.sulekha.com/blog/post/2006/11/galileo-not-einstein-is-inv e ntor-of-second-postulate.htm -------------------------------------- 100 Years of E=mc2 (Book will be published in Dec. 2006 , By NOVA Science, New York, USA) http://physicsajay.sulekha.com/blog/post/2006/11/galileo-not-einstein-is-inv entor-of-second-postulate.htm 1. What is E=mc2? What is its importance? E=mc2 is the most wonderful and significant equation is physics. In 1945 the explosion of atomic bombs on Hiroshima and Nagasaki were base upon this equation. According to this mass (m) can be converted to energy (E) and energy can be converted to mass. EinsteinÍs 27 Sep 1905 paper available at http://www.fourmilab.ch/etexts/einstein/E mc2/www/ 2. This equation is doing well since past 100 years then where is the inconsistency? The inconsistency lies in its mathematical derivation (a method to obtain a mathematical equation). In his 1905 paper Einstein did not derive it mathematically but in true sense speculated it. Einstein earlier derived L = mc2 (light energy mass conversion equation). Then Einstein speculated that what is true for light energy (L) the same is true for every energy (E). This speculation results in E=mc2, such a significant equation must be based upon a specific mathematical derivation and not on speculation. 3. Is EinsteinÍs derivation of L =mc2 correct? The derivation of L=mc2 is incomplete or true in special conditions only. Einstein took just handpicked parameters out of numerous possible, to obtain the equation. Einstein was aware of the reality so he left in midway after getting the desired result. If all valid values of parameters are taken, then results are contradictory in nature. 4. What are contradictory results? Some UNDISCUSSED predictions of EinsteinÍs 29 Sep. 1905 derivation blatantly contradict Law of Conservation of Matter. I have scientifically confirmed the same. No limitation can be bigger than this in science. 5. Was E=mc2 or similar ideas existed before Einstein? Yes, E=mc2 existed before Einstein. An Italian Olinto de Pretto published E=mc2 in valid scientific journal Lettere ed Atti, Feb. 1904, two years before Einstein. But Pretto died in 1921, before its experimental confirmation in nuclear physics. 6. Einstein speculated E=mc2 from L=mc2. What is the problem here? Firstly derivation of L=mc2 is incomplete or under special conditions only. For examples there are many variables in EinsteinÍs derivation e.g. number of light waves emitted by body, magnitude of light energy, angle at which light energy is emitted and relative velocity v. Einstein just took handpicked values of variables. If general values of variables are taken then results are contradictory to experiments. Secondly Einstein originated E=mc2 on the basis of speculation only without any conceptual and mathematical basis. Basically Einstein replaced L by E in equation L=mc2 to get E=mc2. 7. Then how did you derive new equation, dE =Ac2dm (or DE = Ac2 DM )? I have derived new equation between mass-energy conservation by simple calculus method. In dE =Ac2dm, A is a co-efficient of proportionality like numerous others in science. It is dimensionless variable. http://www.burningbrain.org/pdf/ajaysharma einstein.pdf 8. How do you compare these two equations? Firstly dE =Ac2dm is based upon a conceptual and mathematical derivation. On the other hand E=mc2 is a speculation, it is bitter truth. Secondly dE =Ac2dm is a general equation and E=mc2 is its special case. Energy emitted by new equation can be less, equal to or more than predicted by E=mc2. 9. How did you justify your equation experimentally? In Nuclear Physics there are some anomalous results which cannot be explained by E=mc2 . Like this there are some instances in astrophysics where my equation is extremely useful. 10. Is your work recognized by international scientific community? Yes, it is completely recognized, as published in peer review journals. 11 Have you got any recognition certificate from the scientific community? The only way to get scientific recognition is that to get the work published in peer review international journals and conferences. My research papers are either published in international journals from America, England and Canada or being published. I have got invitation from at least 55 International Conferences to present my work. I have presented my research in international conferences in USA, England, Germany, Taiwan Ukraine etc. I have invitation from France and Italy to present my work this year. [b]Still there ANYONE is welcome to COMMENT on the work in Physics Essays giving the facts e.g. What is EinsteinÍs Sep 1905 paper ? What are conditions under which it is derived? Under what conditions experimentally it holds good? How to generalize it under all conditions? What is Ajay SharmaÍs Interpretation? How Ajay SharmaÍs paper is different from EinsteinÍs Sep 1905 paper How Ajay SharmaÍs interpretation is incorrect (if it)? What are the correct interpretations? I HAVE ANSWERS TO ALL QUESTIONS. If Editor Physics Essays and his Editorial Board finds your interpretation and published the paper it is OK. All the references are given below for the purpose. There may be back door critics but none of the scientists have dared to write to scientific bodies or journals Editors , that Ajay SharmaÍs work is incorrectly published. 12. Can this work be introduced in Schools and colleges? Yes my wok is scientifically approved in journal in USA, CANADA and England. Hence it can be so done by any country. IT IS THE IMPORTANCE OF THE WORK. 13. How do you counter the opposition of the people which has come in you your way? Science is the international language. For this, I take seriously the logical conclusions of the critics. I completely ignore the irresponsible critics, as they donÍt exist. The critics when understand the things become my supporters. 14. What about your book, 100 Years of E=mc2? This book will be published in Dec. 2006 It will bring clear and unbiased picture of the facts. The contents of book are already approved by expert scientists after PEER REVIEW and published in international journals and conferences .The book is meant for general public who is interested in basic science. This book will status as NewtonÍs Principia or GalileoÍs Dialogue have in science. Book Link : https://www.novapublishers.com/catalog/product info.php?cPath=23 48 324&products id=4554 Interviewer Rajesh Thakoor Email mc2.book@gmail.com References of EinsteinÍs work . A.Einstein, Annalen der Physik 18 (1905) 639-641. . DOES THE INERTIA OF A BODY DEPEND UPON ITS ENERGY-CONTENT? Weblink is EinsteinÍs 27 Sep 1905 paper available at http://www.fourmilab.ch/etexts/einstein/E mc2/www/ PartII References of Ajay SharmaÍs work My work is available at îThe Origin of Generalized Mass-Energy Equation ?E = Ac2 ?M; and its applications in General physics and Cosmologyî. http://www.burningbrain.org/pdf/ajaysharma einstein.pdf For details 100 Years of E=mc2 https://www.novapublishers.com/catalog/product info.php?cPath=23 48 324&products id=4554 International Conferences It has been accepted for presentation over 55 conferences all over the world --------------------------------------few of them 1. Sharma, A. presented in 19th International Conference on the Applications of Accelerators in Research and Industry , 20-25 August , 2006 Fort Worth Texas, USA 2. A. Sharma, Abstract Book 38th European Group of Atomic Systems ( Euro physics Conference) Isachia (Naples) Italy (2006) 53. 3. A. Sharma , Abstract Book , A Century After Einstein Physics 2005 , 10-14 April 2005 ( Organizer Institute of Physics , Bristol ) University of Warwick , ENGLAND 4. A. Sharma presented in 5th British gravity Conference , OXFORD ENGLAND 5. A. Sharma,. Proc. Int. Conf. on Computational Methods in 6. A. Sharma, Proc. Int. Conf. on Number, Time, Relativity United plus more -------------------------------------- Journals This paper îThe Origin of Generalized Mass-Energy Equation ?E = Ac2 ?M; and its applications in General physics and Cosmologyî. is published in journal Physics Essays , CANADA www.physicsessays.com The paper The past, present and future of E=mc2 will be published in 2007 Galilean Electrodynamics, Massachusetts, USA. In parts it is published in various others journals. ---------------------- Book 100 Years of E=mc2 For details https://www.novapublishers.com/catalog/product info.php?cPath=23 48 324&products id=4554 Email ajay.sharmaa@rediffmail.com 0091 94183 09989, 0091 177 2804546 === Subject: Re: History, Mystery and Chemistry of E=mc2. > Wow! This guy has sock puppets coming out the kazoo! > === Subject: Re: History, Mystery and Chemistry of E=mc2. I didn't see E= mc^2 but I will comment on two things that are right at the beginning of the website given. First the assertion Gallileo is inventor of relativity is true, and everyone knows it, except for one thing: most people make a distinction between Einsteinian relativity, which accounted for electro-magnetic phenomena, and Gallilean relativity which does not since Gallileo did not know anything about electro-magnetism. The addition of electro-magnetism is precisely the thing that makes relativity so different from classical physics. Second, the claim to have calculated that, under certain conditions, emitting a light ray can INCREASE mass, contains arithmetic errors. This may be because the author was assuming that the direction is important in the Lorentz contraction- that in one direction the change is positive, in the other negative- and that is not true. === Subject: vectors to create an orthonogal base I have three vectors with blanks, v=(0.8, -0.6, []), u=([],[],1), w=([],0.8,[]). I'm supposed to fill in the blanks so that the vectors create an orthonogal base. Would be thankful for tips and clues rather than a straight answer. === Subject: Re: vectors to create an orthonogal base > I have three vectors with blanks, v=(0.8, -0.6, []), u=([],[],1), > w=([],0.8,[]). I'm supposed to fill in the blanks so that the vectors > create an orthonogal base. > Would be thankful for tips and clues rather than a straight answer. > orthogonal means the dot product of the vectors is 0. so v.u=0, v.w=0,u.w=0. Applying the definition of the dot product, you will get 3 linear equations with 3 unknowns, which you solve via gauss elimination or some other method === Subject: Re: vectors to create an orthonogal base /Johan === Subject: Re: vectors to create an orthogonal base days. My association with the Department is that of an alumnus. >I have three vectors with blanks, v=(0.8, -0.6, []), u=([],[],1), >w=([],0.8,[]). I'm supposed to fill in the blanks so that the vectors >create an orthonogal base. I assume you mean orthogonal. I fixed the subject as well. With respect to what inner product? > Would be thankful for tips and clues rather than a straight answer. I assume the usual (dot) product. Write v = (0.8, -0.6, a) u = (b, c, 1) w = (d, 0.8, e). Since =0, we have 0 = <(0.8,-0.6,a),(b,c,1)> = a + 0.8 b - 0.6c 0 = bd + 0.8 c + e and from =0 you get 0 = 0.8d +ae - 0.48 Try solving for a, b, c, d, and e. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === Subject: A n is a normal subgroup of Sn I need help? 1.How to explain A n is a normal subgroup of Sn? 2. In S4, find a cyclic subgroup of order 4 and non-cyclic subgroup of order 4? 3. Give an element of A8 whose order is 15. Thx a lot === Subject: Re: A n is a normal subgroup of Sn > I need help? > 1.How to explain A n is a normal subgroup of Sn? Very quick proof: the index of this subgroup is 2. Every subgroup of index 2 is normal. If that's not easy for you to prove, you're missing something in the definition of normal subgroup or something about the partition of the whole group into cosets. -- Mike Hardy === Subject: Re: A n is a normal subgroup of Sn days. My association with the Department is that of an alumnus. >I need help? >1.How to explain A n is a normal subgroup of Sn? Show it is invariant under conjugation. Which is easy: (i) the parity of a permutation is the same as the parity of its inverse. (ii) What is the parity of g^{-1}ag for any permutation g and a of S_n? (iii) What if a is in A_n? >2. In S4, find a cyclic subgroup of order 4 and non-cyclic subgroup of order 4? Ehr... Find an element of S_4 of order 4? Find a subgroup of order 4 that does not have any elements of order 4 (hence all elements are of order 2)? >3. Give an element of A8 whose order is 15. Don't you think you should be doing your own homework? -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === Subject: cancel Control: cancel This message was cancelled from within Mozilla. === Subject: Re: help with inverse laplace transform > i have a problem where F(s) = 1 / s (s^2 + s + 1) and > cant figure out how to use laplace transform to solve > this for f(s). any help would be appreciated You also might want to complete the square: s^2+ s+ 1= s^2+ s+ 1/4+ 3/4= (s+ 1/2)^2+ 3/4 You can use partial fractions to write 1/(s(s^2+ s+ 1)) as A/s+ (Bx+ C)/((s+1/2)^2+ 3/4) === Subject: Re: please help simple group theory > Sounds good. > But wait. > The problem was written as > In Z_^, the group of integers modulo 6, find the > order of each element. How do you know that the > operation they wanted us to consider was binary > addition and not binary multiplication? A really good way is to read the DEFINITION of the group of integers modulo 6!! === Subject: 2d curves to 3d surfaces Hello to one and all and all and one. I am currently underway with a project to model the magnetic connection time of the cluster satellites to the terrestrial bow shock, sounds fun doesn't it? My first point of call is to aquire the equation for the 3D surface that is the terrestrial bowshock (best thought of as a half of an ellipsoid type shape). I have a specific model to use for the bowshock derived by a bloke called Farris which is in 2D and in polar coordinates, my equation needs to be in 3D cartesian coodinates. I currently have a cartesian representation in 2D: x = (2*k*Eps - sqrt( (-2*k*Eps)^2-4*(Eps-1)*(k^2 - y^2) ) / 2*(Eps - 1) where y is between -50 and 50, k is a constant related to solar wind ram pressure and Eps is the eccentricity of the ellipsoid. So now for my question: I need the equation of the surface obtained by rotating this curve about the x axis in essence creating a paraboloid/half an ellipsoid, can anybody please help!? I get the feeling this is very simple to do and I've just not been looking in the right places. PETE === Subject: Re: 2d curves to 3d surfaces On 6 Nov 2006 14:15:25 -0800, Peter b I am currently underway with a project to model the magnetic connection >time of the cluster satellites to the terrestrial bow shock, sounds fun >doesn't it? >My first point of call is to aquire the equation for the 3D surface >that is the terrestrial bowshock (best thought of as a half of an >ellipsoid type shape). I have a specific model to use for the bowshock >derived by a bloke called Farris which is in 2D and in polar >coordinates, my equation needs to be in 3D cartesian coodinates. >I currently have a cartesian representation in 2D: x = (2*k*Eps - sqrt( (-2*k*Eps)^2-4*(Eps-1)*(k^2 - y^2) ) / 2*(Eps - 1) For starters, your parentheses are unbalanced so you need to give the corrected equation. --Lynn === Subject: Re: 2d curves to 3d surfaces Hello to one and all and all and one. >I am currently underway with a project to model the magnetic connection >time of the cluster satellites to the terrestrial bow shock, sounds fun >doesn't it? >My first point of call is to aquire the equation for the 3D surface >that is the terrestrial bowshock (best thought of as a half of an >ellipsoid type shape). I have a specific model to use for the bowshock >derived by a bloke called Farris which is in 2D and in polar >coordinates, my equation needs to be in 3D cartesian coodinates. >I currently have a cartesian representation in 2D: x = (2*k*Eps - sqrt( (-2*k*Eps)^2-4*(Eps-1)*(k^2 - y^2) ) / 2*(Eps - 1) For starters, your parentheses are unbalanced so you need to give the > corrected equation. --Lynn My utmost appologies, late night and all that, it's simply the quadratic formula: (-b - sqrt(b^2 - 4*a*c))/(2*a) (-(-2*k*eps) - sqrt( (-2*k*eps)^2 - 4*(eps-1)*(k^2 - yCoords.^2))) / (2*(eps-1)); Im not quite so bothered about an exact answer but rather just the method involve in obtaining the exact answer, but any help would be much apreciated. Peter === Subject: Re: 2d curves to 3d surfaces On 7 Nov 2006 00:24:00 -0800, Peter b > On 6 Nov 2006 14:15:25 -0800, Peter b Hello to one and all and all and one. >>I am currently underway with a project to model the magnetic connection >>time of the cluster satellites to the terrestrial bow shock, sounds fun >>doesn't it? >>My first point of call is to aquire the equation for the 3D surface >>that is the terrestrial bowshock (best thought of as a half of an >>ellipsoid type shape). I have a specific model to use for the bowshock >>derived by a bloke called Farris which is in 2D and in polar >>coordinates, my equation needs to be in 3D cartesian coodinates. >>I currently have a cartesian representation in 2D: >>x = (2*k*Eps - sqrt( (-2*k*Eps)^2-4*(Eps-1)*(k^2 - y^2) ) / 2*(Eps - 1) >> For starters, your parentheses are unbalanced so you need to give the >> corrected equation. >> --Lynn >My utmost appologies, late night and all that, it's simply the >quadratic formula: >(-b - sqrt(b^2 - 4*a*c))/(2*a) (-(-2*k*eps) - sqrt( (-2*k*eps)^2 - 4*(eps-1)*(k^2 - yCoords.^2))) / >(2*(eps-1)); Im not quite so bothered about an exact answer but rather just the >method involve in obtaining the exact answer, but any help would be >much apreciated. >Peter Here's a link to my old hand-written class notes from teaching calculus. Look at Lecture 27 marked page 156 for a worked example from which you can learn the method. http://math.asu.edu/~kurtz/LectureNotes/ --Lynn === Subject: Re: series convergence Mon, 6 Nov 2006 02:15:45 -0000 from Mike Terry : > William's statement is better as it allows the minus signs to be inserted > arbitrarily into the series. Your statement only talks about inserting the > minus signs at every even numbered term, or at every odd numbered term I wrongly ASSumed the OP was talking about alternating series, but of course (sin n)(cos n)/9^n is not an alternating series. And anyway, as you say, William's statement is more general. me. :-) -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com/ === Subject: Re: series convergence On Sun, 5 Nov 2006 18:34:10 -0800, William Elliot >> : > if sum(n=0,oo) a_n converges absolutely, >> ie if sum(n=0,oo) |a_n| converges >> then sum(n=0,oo) a_n converges. >> I see no difference between what I said and what you said. >> William's statement is better as it allows the minus signs to be >> inserted arbitrarily into the series. Enough of this quibbling. >What's a simple proof that absolute convergence implies convergence? Big Hint: By definition a series converges if and only if the _sequence_ of partial sums converges. That happens if and only if the sequence of partial sums is a Cauchy sequence... ************************ David C. Ullrich === Subject: Re: series convergence <6cesk2d51bjpsp26vk9g0rr3telfrt048q@4ax.com> <454e9ad2$0$8747$ed2619ec@ptn-nntp-reader02.plus.net> <4uduk2dmt31juqhbtm2dekhaq7h2mhl5sd@4ax.com >What's a simple proof that absolute convergence implies convergence? Big Hint: By definition a series converges if and only if the _sequence_ > of partial sums converges. That happens if and only if the > sequence of partial sums is a Cauchy sequence... > So the absolute value of the difference of two partial sums <= the tail of the absolute value series starting from the end of the shortest partial sum. Hey, can I take a hint? ;) === Subject: Re: series convergence The directions are to Use the direct comparison test to determine if the >series converges or diverges. the problem is sum(1-->inf) (sin n * cos n) / 9^n. My problem arises when I realized that for the comparison test, a series >with all positive terms is required. >> Really? _Exactly_ how is the comparison test stated in your book? In my book, the statement of the Comparison Test is: Suppose that sum(a_sub n) and sum(b_sub n) are series with postive terms. >(i) If sum(b_sub n) is convergent, and sum(a_sub n) <= sum(b_sub n) for all Of course that was a typo for a_sub n <= b_sub n, right? (Note for future reference that your a_sub n is usually written a_n around here...) >n, then sum(a_sub n) is also convergent. >(ii) reads similarly but is for divergent case. Is this a possibly incomplete statement of the test? Well, the theorem is often stated differently, requiring only b_n >= 0 and |a_n| <= b_n instead of b_n >= 0, a_n >= 0 and a_n <= b_n. Of course if you know that an absolutely convergent series converges then you see the two versions are equivalent. >So if I use the (sin n*cos n) / 9^n and 1 / 9^n as a(sub n) and b(sub n) >respectively, can I say the series is convergent using the comparison test >even though a alternates signs periodically? Dustin >> (In most statement _one_ of the series has to be positive. >> Note that 1/9^n > 0.) > Is there a way to use the comparison >test to determine convergence or divergence, if not what method would you >use? Dustin > ************************ >> David C. Ullrich > ************************ David C. Ullrich === Subject: Re: non-euclidean question > I have two problems that I dont know how to solve in a non-euclidean > geometry. > The first is > Given triangles ABC and DEF where angle A = angle D, Angle B = angle E > and segment AC = DF > prove that these two triangles are congruent. Doing this in the > euclidean world is pretty straight forward, but I'm not sure what to > do or look out for in the non-euclidean world. This theorem is Euclid I.26. As Euclid doesn't use his parallel postulate until I.29, his proof of I.26 is absolute, which means that it works unchanged in the hyperbolic plane. > > the second is > Given segments AB and AC, C is distinct from B, and segments EC and > DB, where E is on segment AB and segment EC is perpendicular to AB > and D is on segment AC and segment BD is perpendicular to segment AC. > Show that if segment BD = CE then AB = AC. > This I was also able to do in the euclidean world but not in the > non-euclidean. The shortest proof I know says that triangles BCD, CBE are congruent by the RHS criterion, i.e. using a right angle, hypotenuse and another side. Then angle BCD = angle CBE, so AB = AC by Euclid I.6. Euclid doesn't state RHS congruence as a separate theorem, but many geometry text-books have an absolute proof which copies one of the triangles alongside the other. If you can't find that proof, post again. > > also any suggestions on what does not work in a non-euclidean geometry > would be helpful. Experience helps. You've made a good start by thinking about these examples. Text-books will suggest others. Ken Pledger. === Subject: Re: non-euclidean question > I have two problems that I dont know how to solve in > a non-euclidean > geometry. > The first is > Given triangles ABC and DEF where angle A = angle D, > Angle B = angle E > and segment AC = DF > prove that these two triangles are congruent. Doing > this in the > euclidean world is pretty straight forward, but I'm > not sure what to > do or look out for in the non-euclidean world. The proof in the Euclidean world is based on the fact that the sum of angles in a triangle is 2 right angles. That is not true in the non-Euclidean world and the theorem itself is not true. > > the second is > Given segments AB and AC, C is distinct from B, and > segments EC and > DB, where E is on segment AB and segment EC is > perpendicular to AB > and D is on segment AC and segment BD is > perpendicular to segment AC. > Show that if segment BD = CE then AB = AC. > This I was also able to do in the euclidean world but > not in the > non-euclidean. > > also any suggestions on what does not work in a > non-euclidean geometry > would be helpful. === Subject: limit problem How to do this problem: Sequence (x_n) satisfies: 7x_{n+1} = (x_n)^3 + 6 for n >= 1. If x_1 = 1/2, show the sequence is increasing and find its limit. === Subject: Re: limit problem > How to do this problem: > > Sequence (x_n) satisfies: 7x_{n+1} = (x_n)^3 + 6 for > n >= 1. If x_1 = 1/2, show the sequence is > increasing and find its limit. I would prefer to write the recursion as x_{n+1}= (x_n)^3/7+ 6/7. Use induction on n. x_1= 1/2 and x_2= (1/2)^3/7+ 6/7= 1/56+ 48/56= 49/56= 7/9> 1/2 so the base case is true. Assume x_k> x_{k-1}. Then x_{k+1}= (x_k)^3/7+ 6/7. Since x_k> x_{k-1}, that is > (x_{k-1}^3/7+ 6/7= x_k and we are done. Of course, to use monotone sequences to show that there IS a limit you will now need to show that this sequence has an upper bound. 1 should work. If there does exist a limit, say a, then taking limits on both sides of 7x_{n+1}= (x_n)^3+ 6 gives 7a= a^3+ 6. === Subject: Re: limit problem > How to do this problem: Sequence (x_n) satisfies: 7x_{n+1} = (x_n)^3 + 6 for n >= 1. If x_1 = > 1/2, show the sequence is increasing and find its limit. > Clearly (prove) by induction (x_n)_n is increasing whenever x_1 >= 0. (x_n)_n is bounded above by 1 whenever 0 <= x_1 <= 1 Thus by theorem a = lim(n->oo) x_n exists Now apply theorems about limits and sums to show 7a = a^3 + 6 But don't stop now, there's more (for you) to do. === Subject: Different ways to attack a proof As it sometimes happens in math studies I'm stuck with a proof. Instead of stating the problem here (it is quite long) I thought I'll ask general methods people use when stuck. So, do you have some special ways to get over a hard proof? Perhaps starting from the result instead of the premises or definitions? Is brute force sometimes a good method? Something else, maybe a short walk? Where do you get the ideas? I'm looking for some general methods which I could deploy. It certainly looks that I'm not getting over my proof without some new ideas. === Subject: Re: Different ways to attack a proof > As it sometimes happens in math studies I'm stuck with a proof. Instead > of stating the problem here (it is quite long) I thought I'll ask > general methods people use when stuck. So, do you have some special ways to get over a hard proof? Perhaps > starting from the result instead of the premises or definitions? Is > brute force sometimes a good method? Something else, maybe a short > walk? Where do you get the ideas? I'm looking for some general methods which I could deploy. It certainly > looks that I'm not getting over my proof without some new ideas. > In addition to the Polya classic, you might also enjoy reading and using How to Read and Do Proofs: An Introduction to Mathematical Thought Processes, by Daniel Solow. But remember that while there may be models which can help you do proofs , sometimes you must use more than one model in a complicated proof. Never forget that it took more than 300 years to prove Fermat's theorem. Bernard Mass.8e === Subject: Re: Different ways to attack a proof > As it sometimes happens in math studies I'm stuck with a proof. Instead > of stating the problem here (it is quite long) I thought I'll ask > general methods people use when stuck. > > So, do you have some special ways to get over a hard proof? Perhaps > starting from the result instead of the premises or definitions? Is > brute force sometimes a good method? Something else, maybe a short > walk? Where do you get the ideas? > > I'm looking for some general methods which I could deploy. It certainly > looks that I'm not getting over my proof without some new ideas. A very good question. There are some very good answers in the little book by G. Polya, How to Solve it. You can learn a lot just from his list on pp. xvi-xvii, explained on pp. 1-36. Ken Pledger. === Subject: Re: Different ways to attack a proof > As it sometimes happens in math studies I'm stuck with a proof. Instead > of stating the problem here (it is quite long) I thought I'll ask > general methods people use when stuck. > > So, do you have some special ways to get over a hard proof? Perhaps > starting from the result instead of the premises or definitions? Is > brute force sometimes a good method? Something else, maybe a short > walk? Where do you get the ideas? > > I'm looking for some general methods which I could deploy. It certainly > looks that I'm not getting over my proof without some new ideas. I highly recommend the following book: http://en.wikipedia.org/wiki/How_to_Solve_It -- -kira === Subject: Re: Different ways to attack a proof I will often try doing some calculations with examples to see why they work (or where they fail). For example- we had to recently prove a fact about factor groups. So I picked some (small) example groups and picked a normal subgroup to for the factor groups with - since the proof specified normal subgroups, I also looked at what would happen if I tried it with a non-normal subgroup. Why did the proposition fail? When it worked, what made things come together? Is there a pattern? I also sometimes try looking at the converse or contrapositive of the proposition to see if that might be easier to prove (or disprove) - that sometimes sheds light on how to go about attacking a proof. If you are doing homework problems, you might also look into how similar proofs were done earlier in the chapter - what techniques were used. Be warned that this can be misleading! The chapter might highlight some proof technique (say, induction), but the particular problem you are working on might be more easily solved with a proof by contradiction. Hope that helps... Eric === Subject: Fractal Forums - Image contest - Win a DVD! Main competition URL: http://www.fractalforums.com/index.php?board=81.0 BIOCURSION DVD COMPETITION An exciting co-venture between Fractal Forums (Cambridge, UK) and Pixel Mavel Productions (Seattle, USA) We are giving away 3 copies of Pixel Maven Production's Biocursion DVD, a 120 minute compilation of fractal animations with specially commissioned sountracks, as prizes in competitions on Fractal Forums. The competitions are open to Fractal Forums users to generate imagery and animations in the following categories: * Best Fractal Forums Logo * Best Static Image * Best Fractal Animation The competitions run concurrently until the END OF NOVEMBER so get your skates on and enter! About BIOCURSION (www.biocursion.com): biocursion is a full-length film of fractal animation, boasting a play time of 122 minutes. This film has been in development stages for several years; a montage of approximately twenty-five fractal animations set to a new age, ambient and techno soundtrack. Being fans of fractal art, we found a void which needed to be filled - and that void was to see fractal animation taken to the next level.. biocursion includes the works of four fractal animators, including two Jock Cooper and Lloyd Garrick (Jack of Tradez). The primary contributors of music are Paul Kay and Scott Wade. Read more about the Biocursion DVD http://www.biocursion.com About Fractal Forums (www.fractalforums.com): FractalForums.com is a forum for Fractal related discussion. Share your own fractal images and fractal movies with us. Discuss fractal theory, IFS, Mandelbrot sets, fractal art and even stock market prediction! Some of our users are teachers and professors - some are high school students - some come along just for the ride - but all share an enthusiasm for all things Fractal related. A phenomenal first month's growth at 140 new members and nearly 1200 posts. === Subject: A huge problem given R = 1 + 1/2 + 1/3 + 1/4 + .... + 1/n + .... the sum does not exist.... given f(x) = 1 / x f(x) Integrated from 0 to infinity gives Lim (x-->infinity) ln x = infinity so the sum (of areas) does not exist can anyone make sense of this??? this is the best I got: because they are approching 0 too slow, they do not converge.. is is not a very satisfying answer.. to make the problem clear: how can something that has 0 as the limit of the nth number not approach a finite sum? === Subject: Re: A huge problem > given R = 1 + 1/2 + 1/3 + 1/4 + .... + 1/n + .... > > the sum does not exist.... So you really shouldn't write R = - a picky point. > given f(x) = 1 / x > f(x) Integrated from 0 to infinity gives Lim (x-->infinity) ln x = > infinity > so the sum (of areas) does not exist > > can anyone make sense of this??? > > this is the best I got: because they are approching 0 too slow, they do > not converge.. > > is is not a very satisfying answer.. > > to make the problem clear: how can something that has 0 as the limit of > the nth number not approach a finite sum? You're surprised by the wrong thing. The surprise is that any series with postitive terms converges. After all, the partial sums just get bigger and bigger and bigger and ... The best you've got is probably the best you'll get. For the harmonic series, the partial sums get larger and the terms get smaller but not small enough fast enough. In fact, lim(1 + 1/2 + ... + 1/k - ln(n),n = 1 ... oo) = E where E is the Euler-Mascheroni constant (a little bit larger than 1/2). Although it would seem to be the same situation, 1 + 1/2 + 1/9 + ... + 1/n^2 + ... converges. The partial sums get larger but only increase by 1/n^2 at each step rather than by 1/n. I'm sure you have studied or will study p-series. -- Paul Sperry Columbia, SC (USA) === Subject: Re: A huge problem On 7 Nov 2006 14:13:57 -0800, Fallingeagle >given R = 1 + 1/2 + 1/3 + 1/4 + .... + 1/n + .... the sum does not exist.... given f(x) = 1 / x >f(x) Integrated from 0 to infinity gives Lim (x-->infinity) ln x = >infinity >so the sum (of areas) does not exist can anyone make sense of this??? this is the best I got: because they are approching 0 too slow, they do >not converge.. is is not a very satisfying answer.. to make the problem clear: how can something that has 0 as the limit of >the nth number not approach a finite sum? It just can. Nothing says you have to like it. Think about this sum: 1 + 1/2 + 1/2 + 1/3 + 1/3 + 1/3 + 1/4 + 1/4 + 1/4 + 1/4 + ... where there are n terms of 1/n at each stage. --Lynn === Subject: Re: A huge problem Here is another standard example: 2*(sqrt(n+1)-1)1 and diverges if d<=1. === Subject: Is Katherine Trimble an envy racist bigot? Is Katherine Trimble an envy racist bigot? A person, mistakenly think that she's in his friend list. However, rather than denying it, she'll simply give a friendly reply. >Hi [My Friends' name] >Remind me what *** is again? There are some networks that I simply don't bother with anymore. Best wishes Katherine Such mutual communication automatically make each person in the friend list of another in a social networking site. Not suspicious of any problem, the person then build his business network in a friendly manner to many of those in his list, which unfortunately include the old hag. Soon, the beldam is engaged in serious slandering campaign the person. She claimed that never communicated at all with the person, which is false. She claimed that the person is spamming, the claim that does not match the official rule. She also claimed that the person is pretending to be her friend to get her friends. The accusation is absurd. Who the hell does she think she is that anyone would pretend to be her friend? What makes her so arrogant? What sort of idiot cannot tell a different between someone trying to be friendly to many people and someone trying to fraudulently get extra credibility by pretending to be someone's friend? Thinking this is just a friendly misunderstanding, the person simply explained to Katherine: >I don't mind your comments about me as long as it is true. So keep in mind that we did chat in a friendly way and we were indeed in >each other network, which you can check, before saying anything about me. Also, *******'s definition of spamming is trying to sell >something without intent to network which is far from what I do. >I'll try to get both of us disconnected. ...you could have understood and tell me privately like Peter did. Then we could have really been >friend. It's a tragedy, but I got 400+ other friends to concentrate on. The beldam didn't stop. She got even madder. She kept making many false claims. She claimed that the person is a robot with no public information. She said that she had never communicated in any way to the person. She tried to portray that the person has some fraudulent intent, friendless, and that nobody trust the person. She claimed that the person is a nazi. All these are over a few friendly gestures to her friends. Now, put your self in the persons' shoe. What can he do to make her leave him alone? Expert analysis suggests racism as her real motives. The person is Asian and Katherine is white. She kept doing so after being told about the fallacy of the claim. It's not until the truth is published that her tone start changing. Then she raised totally different issues. That didn't left not many amicable options to deal with this kind. Guess she knew all along that the claim were false. She just didn't expect anyone would publish the contrary. She's a time bomb. She had gone the extra miles condemning a person for a few slight friendly gesture, we'll never know what she'll do to anyone. It seems that her other much related real motive is best explained by her very own words: I do not like the way you ramp up your connections It's quite unfortunate that the world is filled with envy vermins that just get this thing against those more successful than them. The truth is they're simply uncompetitive. Hence, they craft lies and prejudices against those who are. That's why we have so many nonsenses justifying prohibition against so many consensual acts. Some have reasonable cases. Most can be appeased. But some, like the beldam, should get the out of the gene pool for good along with all her kind for the sake of prosperity for all before they form another Nazi party. It's very stupid trying to reason to such bigots for the same reason we don't reason or negotiate with mosquitoes and germs. We just get rid them. More effortlessly we should set one of the vermin as a sample to show the rest their proper place. They hit, we hit harder; Those maggots don't deserver higher level of communication. Currently our world is filled with unfair misery given to those who are honestly successful. The mere acts of making honest money are punishable by tax. Getting many chicks is also condemned. Yet, we have to lock our doors in fear that someone will steal our properties because governments protect thieves. Unless those who run the fastest also hit the hardest, success will then simply be a bridge to gas chamber. Even if we let the worthless die and kill parasites, the market will take care of everything anyway. Believing otherwise is practicing envy bigotry. Yet, capitalists, out of benevolence have given more and more to those who oppress them. Only capitalists patiently give the other cheek and repay hatred and genocide with aids and helps often to ungrateful angry mobs. When vermins like Katherine keeps being influential, it should be about time all of us to stop playing nice. Keep doing otherwise means having to realize that it's really fear and not benevolent or mercy that motivate us. Long live freedom, competition, rationality, meritocracy, prosperity, and proper alignment between individuals' interests to productivity as a whole. The world will be a better place if all of her kind went extinct. What do you think? === Subject: Re: continuous progressive tax formula Nevermind, I got the answer on Yahoo!Answers from Wal C: Tax = B[income - threshold] + R[income - threshold]^P where B is base (minimum) tax rate (e.g., 0.30) R is what he calls the power tax rate (e.g., 0.10) and P, an exponent, is power tax index (e.g., 1.05) You can play with the values to get what you want. === Subject: Re: Help! Equation for Length of Coiled Wire? what do you mean by length of part of cylinder that was wrapped