mm-215 === Subject: by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1R3FAX19599; by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1R36pi18853 by legacy.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: === what?Wle I applaud your impulse to be polite, it's counterproductive to start a new thread with it.-- Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.comAn expense does not have to be required to be considered necessary. -- IRS Form 1040 line 23 === impulse to be polite, it's counterproductive> to start a new thread with it.For Christ's sake, Stan, end your ßaming rampage and either ignore it orsay sometng constructive. Not everyone will meet your standard, but youdon't have to remind us of each and every instance. The signal-to-noiseratio of your posts is amazing. Such is notng new from you.And yes, if you would only *consider* to stop ßaming people, my suggestionto you will be constructive. Just let it === problem with ts solution: n-1I(n) = SIGMA Floor[ SQRT(n^2 - i^2) ] i=1How do I create the funtion I(n) so I can directly calculate ts functionwithout doing all the summations? I can approximate it with polynomialinterpolation and write a program to do the calcs faster but would like toknow if an exact formula can be found.How do I go about === solved a problem with ts solution:> n-1> I(n) = SIGMA Floor[ SQRT(n^2 - i^2) ]> i=1> How do I create the funtion I(n) so I can directly calculate ts> function without doing all the summations? I can approximate it with> polynomial interpolation and write a program to do the calcs faster but> would like to know if an exact formula can be found.Your I(n) appears to beNumber of cells of square lattice of edge 1/n inside quadrant of unitcircle centered at 0.If an exact formula were known, it would hopefully have been listed === formula.>I have solved a problem with ts solution:> n-1>I(n) = SIGMA Floor[ SQRT(n^2 - i^2) ]> i=1>How do I create the funtion I(n) so I can directly calculate ts function>without doing all the summations? I can approximate it with polynomial>interpolation and write a program to do the calcs faster but would like to>know if an exact formula can be found.I don't tnk there is any plausible exact formula. I don't eventnk there is a known exact formula if you remove the Floor from ts.But ts is fairly reasonably approximated over a wide range of n by aquadratic equation, with or without the === recommend any math related REUs that are still considering student applications? In case it changes matters, I'm a minority junior Math/EE double major.,Alex-- Our paper became a monograph. When we had completed the details, wewhy. Ts is === pretty simple, but for some reason I'm getting a differentanswer than that given by both the book and by Maple.int(1/sqrt(9*x^2+12*x-5)Okay. Complete the square on the bottom for 1/sqrt((3*x+2)^2-9)), andsubstitute u = 3*x+2. Then (1/3)du = dx, and it becomes(1/3)int(1/(sqrt(u^2-9))).That shouldn't be hard - sub in u = 3sec(v), dv = 3sec(v)tan(v), andit becomes the very nice (1/3)int(sec(v)) = ln(sec(v)+tan(v)).sec(v) = u/3 and tan(v) = sqrt(u^2-9), so: ln(u/3+sqrt(u^2-9)), wchafter subbing in u = 3*x+2 becomes ln(x+2/3+sqrt(9*x^2+12*x-5)) + c. But the answer is ln(3x+2+sqrt(9*x^2+12*x-5)). I simply cannot seewhy. I'd much === Stupid Integral> Ts is pretty simple, but for some reason I'm getting a different> answer than that given by both the book and by Maple.> int(1/sqrt(9*x^2+12*x-5)> Okay. Complete the square on the bottom for 1/sqrt((3*x+2)^2-9)), and> substitute u = 3*x+2. Then (1/3)du = dx, and it becomes> (1/3)int(1/(sqrt(u^2-9))).> That shouldn't be hard - sub in u = 3sec(v), dv = 3sec(v)tan(v), and> it becomes the very nice (1/3)int(sec(v)) = ln(sec(v)+tan(v)).actually, it's (1/3)int(3*sec(v)) = ln(sec(v) + tan(v)).> sec(v) = u/3 and tan(v) = sqrt(u^2-9), so: ln(u/3+sqrt(u^2-9)), wch ^^^^^^^^^^^No:sec^2(v) = 1 + tan^2(v)u^2/9 = 1 + tan^2(v)tan^2(v) = u^2/9 - 1=> tan(v) = sqrt(u^2/9 - 1)ln(sec(v) + tan(v)) = ln(u/3 + sqrt(u^2/9 - 1)) + c = ln(x + 2/3 + sqrt((9x^2 + 12x + 4)/9 - 1 ) + c = -ln(3) + ln(3x + 2 + sqrt(9x^2 + 12x - 5) ) + c(adding and subtracting ln(3); now redefine c -> c - ln(3).)> But the answer is ln(3x+2+sqrt(9*x^2+12*x-5)). I simply cannot see> why. I'd much appreciate it someone could explain ts to me.-- P.A.C. SmithThe vast majority of Iraqis want to live in a peaceful, free world.And we will find these people and we will bring them === (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1RISib02114; by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1RIQni02038 by legacy.mathforum.org (8.11.6/8.11.6/The Math Forum, experimenting a bit with the possibility of representing numbersby using their prime factorisations. The most straight-forward systemI came up with was ts:2 is written as -any other prime p is written as , where x = (p-1)/2 (write 3 as <>any non-prime is written by juxtaposition of the primes it is composedof (preferably in increasing order).(By convenion, write 1 as <)Ts system gives the following codes for the first 10 numbers:1 <2 -3 <>4 --5 <->6 -<>7 <<>>8 ---9 <><>10 -<->Ts system is clearly unambiguous, because of the bracketing of theprimes.Next I was considering various ways to construct similar, but moreefficient, encoding systems, i.e. systems where the codes don't gettoo long too quickly (without using more symbols).The most promising I came up with goes as follows:2 is written as >3 is written as -any other prime p is written as , where x = (p-1)/2.any non-prime is written by juxtaposition of the primes it is composedof, in increasing order.(By convention, write 1 as <)Ts system gives the following codes for the first 10 numbers:1 <2 >3 -4 >>5 <>>6 >-7 <->8 >9 --10 ><>>Ts system would certainly make the codes of many numbers muchshorter, namely all those that depend on 3, as 3 is now represented bya single symbol, not by two.The only question is: Is the system not ambiguous? For The bracketscan't disambiguate it as in the first system, because I am using > tomean 2 as well.I couldn't find any ambiguous strings, and it seemed to me that onemight be able to prove their aren't any. However, I didn't manage toprove it. My approach for a proof was as follows:Use redactio ad absurdum:If their are ambiguous strings, then there must be a shortest string(or group of strings), wch can be interpreted in various ways.Consider the different cases of what t string might look like, andshow for each case, how ts leads to a contradiction with theassumption that there were no shorter such strings.I managed it for many subcases, but not for all.Maybe I am mistaken, and their is an ambiguous string. A nice way toget an idea of what the answer is would be to use a computer to checkit for the first 10,000 numbers or so. Unfortunately, my knowledge ofcomputing aren't good enough to do such a tng.Does anyone see a way to solve ts problem mathematically? Or does atleast someone know enough about computers to tell me, how one couldcheck it for many === number-code> as 3 is now represented by a single symbol, not by twoCan you explain ts in more detail please? 3 to me is a single digit.KevPS In fact your first 10 offerings seem to represent what is 11 digits with22 symbols.PPS Your === answer>Plug in t=0Please observe Usenet conventions: use the same thread title with Re: in front of it, and give _some_ context (not a full quote) Ts makes discussions much easier to follow, and makes your own contributions more valuable.-- Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.comAn expense does not have to be required to be considered necessary. -- IRS Form 1040 line 23 === ßu epidemic in a small town, a public health offical >found that the total # y of people who have caught the ßue after t>days is approximated by y = -t(squared) + 26t + 106. 0 less than or>equal to t less than or equal to 13.>a.)How many have caught the ßu at the beginning of the survery?What does the beginning of the survey mean, in terms of the two available variables t and y? Is it some particular value of y? If so, plug it in and find t. Is it some particular value of t? If so, plug it in and find y.With respect, just posting your homework problems will get you answers so you can say you've done your homework, but gives you little help in actually learning the material. A MUCH better practice is to post what you have done to try to solve the problem, and exactly where you got stuck.If abstract love of learning does not motivate you (and for many busy students I can understand that it does not), tnk about your upcoming exam: if you can't do these problems on your own, prospects are grim.-- Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.comAn expense does not have to be required to be considered === Re: Linear equations...> 1.) During a ßu epidemic in a small town, a public health offical> found that the total # y of people who have caught the ßue after t> days is approximated by y = -t(squared) + 26t + 106. 0 less than or> equal to t less than or equal to 13.> a.)How many have caught the ßu at the beginning of the survery?> how do i find out the question they asked??Write it like ts: y = -t^2 + 26t + 106 0 <= t <= 13At the beginning t = 0, I suppose, so y = 106.Btw, why === riddle my english teacher has been tellinAnd therefore, naturally, you posted it in a math group.What IS it with all these idiots posting riddles here?-- Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.comAn expense does not have to be required to be considered necessary. -- IRS Form 1040 line 23 === support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1RGRMN22206; by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1RGIti21221 by legacy.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: teacher has been tellin us nd we ave to>figure it out .......here it is......dere is a man in a van nd hes>dead on the bed dere is anuva man outside and he is laugng dere is>also two bits of wood in the van the man in the van commited suicide>why?......dis is wat we no so far......both the men r clowns..the>wood has been sawed into 2 pieces but isnt in half the wood has a>particular purpose to the dead man life and thats y he commited>suicide the man on the outside wanted the dead mans job so he wanted>m to commit suicide .....PLZ HELP ME!!!! Your English teacher? Did your English teacher really use wordslike dere and anuva? The dead man was a heavy smoker. The piece of wood was s onlymatch. The man outside had broken it in two pieces- making sure thatit wasn't into two equal pieces so IT WASN'T A MATCH! Hey, that's about the best your going to get to ts silly tng!What a bunch of === equation r=5 into the rectangular>equationWle there are general methods, probably ts one is easiest to solve by grapng it, looking at it, and writing down the Cartesian equation that corresponds.nt: What shape has a constant radius of 5 units from the center?What is the equation in Cartesian (rectangular) coordinates for that shape?-- Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.comAn expense does not have to be required to be considered necessary. -- IRS Form 1040 line 23 === (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1RDW4L04118; by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id i1R7Eei06125 by legacy.mathforum.org (8.11.6/8.11.6/The Math Forum, convert a polar equation r=5 into the rectangular>equationsince r^2 = x^2+y^2, you get x^2 + y^2 = === identities using de moivre?>, how could i prove sin(a+b)=sin a*cos b + sin b * cos a>or cos(a+b)=cos a*cos b - sina*sinb>using de moivre's formula (cos a + isin a)^n = cos (n*a) + isin(n*a)It's not de Moivre's theorem exactly, but Euler's formula. Seehttp://oakroadsystems.com/twt/sumdiff.htm#sincosAplusmnB( P.S. Contrary to what you may have been told, it is not kewl to post in lower case.)-- Stan Brown, Oak Road Systems, Cortland County, New York, USA http://OakRoadSystems.comAn expense does not have to be required to be considered === could i prove sin(a+b)=sin a*cos b + sin b * cos a> or cos(a+b)=cos a*cos b - sina*sinb> using de moivre's formula (cos a + isin a)^n = cos (n*a) + isin(n*a)>> You don't, use instead>> e^i(a+b) = e^ia * e^ib.>Wch is merely an alternate statement of deMoivre's theorem.> There's quite a bit of confusion here.> First, bravo y (the OP) has been given an impossible task. He> correctly states de Moivre's Theorem wch concerns powers involving a> single angle a. It's not helpful in proving what he wants for two> general angles a and b. Virgil made the same mistake.> Second, William Elliot's formula assumes the OP knows that> e^(ix) = cos(x) + i*sin(x)> The formula he quoted can be written> cos(a + b) + i*sin(a + b) = (cos(a) + i*sin(a))(cos(b) + i*sin(b))> (I wouldn't normally give such detailed help with homework, but> ts time it seemed a good idea to clear the air a bit.)However DeMoive's formula is proven by those trig idenitites,so proving those trig identities by DeMoive's formula is circular.