mm-4209 === Subject: Volume swept by rotating a triangle about a vector I have a problem that I am trying to solve. Suppose that I have a triangle formed by three points A, B, and C. We can assume that the points are in 2D space, i.e. all are on the standard X-Y plane. I also have a 2D vector N that represents an axis of rotation. Now, suppose that I want to find the volume given by rotating the triangle about the axis. One of the points of the triangle will be on the axis. So, the area of the triangle is easy enough to figure out (all praise to Wikipdia: http://en.wikipedia.org/wiki/Triangle). I also know that if I have an arbitrary function f(x) and I want to rotate it around the y-axis, then the volume will be the integral of 2 pi x f(x). However, I'm stuck there. I don't know how to get a suitable f(x) function from my three points so that I can do the integral. So, can anyone point me in the right direction? Dave === Subject: Re: Volume swept by rotating a triangle about a vector On Sat, 16 Jun 2007 19:40:01 -0600, Dave Rudolf triangle formed by three points A, B, and C. We can assume that the >points are in 2D space, i.e. all are on the standard X-Y plane. I also >have a 2D vector N that represents an axis of rotation. Now, suppose >that I want to find the volume given by rotating the triangle about the >axis. One of the points of the triangle will be on the axis. So, the area of the triangle is easy enough to figure out (all praise to >Wikipdia: http://en.wikipedia.org/wiki/Triangle). I also know that if I have an arbitrary function f(x) and I want to >rotate it around the y-axis, then the volume will be the integral of >2 pi x f(x). However, I'm stuck there. I don't know how to get a >suitable f(x) function from my three points so that I can do the >integral. So, can anyone point me in the right direction? >Dave Say your triangle has vertices (x1, y1), (x2, y2) and (x3, y3). Your source above gives the area A of the triangle in terms of those coordinates, so you know A. The centroid of the triangle has the average of the corresponding coordinates for its coordinates: C = ( (x1 + x2 + x3 ) / 3, (y1 + y2 + y3) / 3 ). Pappus' second theorem says the volume generated by rotating a plane area A about an axis is equal to the area times the distance the centroid travels. So the problem is reduced to finding how far the centroid is from the axis of rotation. (Do you know how to do that? It can be done purely with vectors.) Say that distance is d. Then your answer is V = 2 pi d A. No integrals needed. All the above assumes the triangle doesn't overlap itself during the rotation. --Lynn === Subject: Re: Volume swept by rotating a triangle about a vector > On Sat, 16 Jun 2007 19:40:01 -0600, Dave Rudolf Hey all, > I have a problem that I am trying to solve. Suppose that I have a > triangle formed by three points A, B, and C. We can assume that the > points are in 2D space, i.e. all are on the standard X-Y plane. I also > have a 2D vector N that represents an axis of rotation. Now, suppose > that I want to find the volume given by rotating the triangle about the > axis. One of the points of the triangle will be on the axis. > So, the area of the triangle is easy enough to figure out (all praise to > Wikipdia: http://en.wikipedia.org/wiki/Triangle). > I also know that if I have an arbitrary function f(x) and I want to > rotate it around the y-axis, then the volume will be the integral of > 2 pi x f(x). However, I'm stuck there. I don't know how to get a > suitable f(x) function from my three points so that I can do the > integral. So, can anyone point me in the right direction? > Dave Say your triangle has vertices (x1, y1), (x2, y2) and (x3, y3). Your source above gives the area A of the triangle in terms of those > coordinates, so you know A. The centroid of the triangle has the average of the corresponding > coordinates for its coordinates: C = ( (x1 + x2 + x3 ) / 3, (y1 + y2 + y3) / 3 ). Pappus' second theorem says the volume generated by rotating a plane > area A about an axis is equal to the area times the distance the > centroid travels. So the problem is reduced to finding how far the centroid is from the > axis of rotation. (Do you know how to do that? It can be done purely > with vectors.) Say that distance is d. Then your answer is V = 2 pi d A. Ya, I can find the vector that is orthogonal to N, then project the vector to C onto that orthog vec. Then get the length of the projected vector. No integrals needed. All the above assumes the triangle doesn't > overlap itself during the rotation. Uh, the triangle would generally have one point on the rotation axis N. Not sure if that counts as self-overlapping or not. --Lynn === Subject: Re: Volume swept by rotating a triangle about a vector On Sun, 17 Jun 2007 11:26:22 -0600, Dave Rudolf Uh, the triangle would generally have one point on the rotation axis N. >Not sure if that counts as self-overlapping or not. > As long as the axis of rotation doesn't pass through the interior of the triangle you are OK. The triangle needs to be all on one side or at worst just touching the edge of the axis. --Lynn === Subject: Delta symbol in engineering text I have an engineering text (Fluid Mechanics by White) that is using a lower case delta symbol in a way I haven't seen before. For example, when talking about shear it says that (tau) = (delta)(theta)/(delta)t Could someone let me know what the lower case delta symbol means in this case? -Rick === Subject: Re: Delta symbol in engineering text On Sat, 16 Jun 2007 20:09:11 -0700, Rick Giuly I have an engineering text (Fluid Mechanics by White) that is using a >lower case delta symbol in a way I haven't seen before. For example, when talking about shear it says that (tau) = (delta)(theta)/(delta)t Could someone let me know what the lower case delta symbol means in >this case? -Rick Likely a differential, or infinitesimal change in the value of a variable. See here under the heading The lower-case letter ? can be used to denote: === Subject: Re: Delta symbol in engineering text I have an engineering text (Fluid Mechanics by White) that is using a > lower case delta symbol in a way I haven't seen before. For example, when talking about shear it says that (tau) = (delta)(theta)/(delta)t Could someone let me know what the lower case delta symbol means in > this case? -Rick > the lower case delta simply means a change in a variable, a definite change rather than an 'infintesimal' change as 'd' does in calculus. === Subject: Re: Delta symbol in engineering text @hotmail.com>: > I have an engineering text (Fluid Mechanics by White) that is using a > lower case delta symbol in a way I haven't seen before. For example, when talking about shear it says that (tau) = (delta)(theta)/(delta)t the lower case delta simply means a change in a variable, a definite change > rather than an 'infintesimal' change as 'd' does in calculus. Are you quite sure about that? Unless I'm mistaken, you're thinking about an upper-case delta, a triangle. The classic use of that is in the definition of the slope of a straight line, delta-y over delta-x. I believe the OP has an instance of a partial derivative, and those are not deltas but Russian d's. -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com/ === Subject: Re: Delta symbol in engineering text > @hotmail.com>: > I have an engineering text (Fluid Mechanics by White) that is using a > lower case delta symbol in a way I haven't seen before. > For example, when talking about shear it says that > (tau) = (delta)(theta)/(delta)t > the lower case delta simply means a change in a variable, a definite > change > rather than an 'infintesimal' change as 'd' does in calculus. Are you quite sure about that? Unless I'm mistaken, you're thinking > about an upper-case delta, a triangle. The classic use of that is in > the definition of the slope of a straight line, delta-y over delta-x. I believe the OP has an instance of a partial derivative, and those > are not deltas but Russian d's. Maybe... but, the lower case deltas used in partial derivatives are similar looking to those used to represent definite changes in calculus, but they arent the same. (And yes capital deltas are also used to represent changes) Only the OP can answer this one. Are these d's to do with partial derivatives..??? -- > Stan Brown, Oak Road Systems, Tompkins County, New York, USA > http://OakRoadSystems.com/ === Subject: Re: Delta symbol in engineering text <92hdi.933$3j1.450@newsfe6-gui.ntli.net @hotmail.com>: > I have an engineering text (Fluid Mechanics by White) that is using a > lower case delta symbol in a way I haven't seen before. > For example, when talking about shear it says that > (tau) = (delta)(theta)/(delta)t > the lower case delta simply means a change in a variable, a definite > change > rather than an 'infintesimal' change as 'd' does in calculus. Are you quite sure about that? Unless I'm mistaken, you're thinking > about an upper-case delta, a triangle. The classic use of that is in > the definition of the slope of a straight line, delta-y over delta-x. I believe the OP has an instance of a partial derivative, and those > are not deltas but Russian d's. Maybe... but, the lower case deltas used in partial derivatives are similar > looking to those used to represent definite changes in calculus, but they > arent the same. (And yes capital deltas are also used to represent changes) Only the OP can answer this one. Are these d's to do with partial > derivatives..??? -- > Stan Brown, Oak Road Systems, Tompkins County, New York, USA > http://OakRoadSystems.com/ I guess OP stands for original person? Anyways, they are deltas not partial derivative symbols. They don't look like any russian letters as far as I can tell. they are using the lower case delta to mean what calculus books usually use a capital delta for. -rick === Subject: Re: Delta symbol in engineering text > @hotmail.com>: > I have an engineering text (Fluid Mechanics by White) that is using > a > lower case delta symbol in a way I haven't seen before. > For example, when talking about shear it says that > (tau) = (delta)(theta)/(delta)t > the lower case delta simply means a change in a variable, a definite > change > rather than an 'infintesimal' change as 'd' does in calculus. > Are you quite sure about that? Unless I'm mistaken, you're thinking > about an upper-case delta, a triangle. The classic use of that is in > the definition of the slope of a straight line, delta-y over delta-x. > I believe the OP has an instance of a partial derivative, and those > are not deltas but Russian d's. > Maybe... but, the lower case deltas used in partial derivatives are > similar > looking to those used to represent definite changes in calculus, but they > arent the same. (And yes capital deltas are also used to represent > changes) > Only the OP can answer this one. Are these d's to do with partial > derivatives..??? > -- > Stan Brown, Oak Road Systems, Tompkins County, New York, USA > http://OakRoadSystems.com/ > I guess OP stands for original person? Anyways, they are deltas not > partial derivative symbols. They don't look like any russian letters > as far as I can tell. they are using the lower case delta to mean what calculus books > usually use a capital delta for. Both lower and upper case deltas are used to represent definite changes in a variable - its usually the case lower are used to represent very small changes and capital to use any sort of change > -rick > === Subject: Re: Delta symbol in engineering text Both lower and upper case deltas are used to represent definite changes in a > variable - its usually the case lower are used to represent very small > changes and capital to use any sort of change I didn't know that. Do the text books that use that convention make it clear what it means? In the expression f(x + delta x) we have to make sure that x + delta x is in the domain of f . We can be sure it is if x is in an open set which is a subset of f's domain and delta x is small enough. That is so whether that's a lower case delta or an upper case delta. -- Remove antispam and .invalid for e-mail address. He that giveth to the poor lendeth to the Lord, and shall be repaid, said Mrs Fairchild, hastily slipping a shilling into the poor woman's hand. === Subject: Re: Delta symbol in engineering text Frederick Williams in a > variable - its usually the case lower are used to represent very small > changes and capital to use any sort of change I didn't know that. Do the text books that use that convention make it > clear what it means? Not sure what textbooks you mean, but throughout my mathematics education lower case delta has always meant an 'infintesimal' change (ie. a very very small change but one still greater than zero), whereas upper case delta has meant any sort of change infintesimal or not. In the expression f(x + delta x) we have to make sure that x + delta > x is in the domain of f . We can be sure it is if x is in an open > set which is a subset of f's domain and delta x is small enough. > That is so whether that's a lower case delta or an upper case delta. -- > Remove antispam and .invalid for e-mail address. > He that giveth to the poor lendeth to the Lord, and shall be repaid, > said Mrs Fairchild, hastily slipping a shilling into the poor woman's > hand. === Subject: Re: Delta symbol in engineering text ' reply-type=response > Maybe... but, the lower case deltas used in partial derivatives are > similar looking to those used to represent definite changes in calculus, > but they arent the same. (And yes capital deltas are also used to > represent changes) If you're talking partial derivatives, don't ask me what they are but I know they are not lower case deltas. Both upper and lower case delta are used for change, depending on context. For ex. the definition of slope of a line, and the definition of limit. -- Darrell === Subject: Re: Delta symbol in engineering text > Maybe... but, the lower case deltas used in partial derivatives are > similar looking to those used to represent definite changes in calculus, > but they arent the same. (And yes capital deltas are also used to > represent changes) If you're talking partial derivatives, don't ask me what they are but I > know they are not lower case deltas. I know they are not lower case deltas... partial derivatives slope to the left... Both upper and lower case delta are used for change, depending on context. > For ex. the definition of slope of a line, and the definition of limit. -- > Darrell > === Subject: Re: Delta symbol in engineering text ' reply-type=response Maybe... but, the lower case deltas used in partial derivatives are > similar looking to those used to represent definite changes in calculus, > but they arent the same. (And yes capital deltas are also used to > represent changes) > If you're talking partial derivatives, don't ask me what they are but I > know they are not lower case deltas. I know they are not lower case deltas... partial derivatives slope to the > left... I was just making a correction since you did say the lower case deltas used in partial derivatives... :-) -- Darrell === Subject: Re: Delta symbol in engineering text @hotmail.com>: > I have an engineering text (Fluid Mechanics by White) that is using a lower case delta symbol in a way I haven't seen before. > For example, when talking about shear it says that > (tau) = (delta)(theta)/(delta)t the lower case delta simply means a change in a variable, a definite change > rather than an 'infintesimal' change as 'd' does in calculus. Are you quite sure about that? Unless I'm mistaken, you're thinking > about an upper-case delta, a triangle. The classic use of that is in > the definition of the slope of a straight line, delta-y over delta-x. Where I come from, small and large deltas are used in that sense. > I believe the OP has an instance of a partial derivative, and those > are not deltas but Russian d's. -- Remove antispam and .invalid for e-mail address. He that giveth to the poor lendeth to the Lord, and shall be repaid, said Mrs Fairchild, hastily slipping a shilling into the poor woman's hand. === Subject: Solutions Manual to Managerial Accounting (Garrison) I would like to have the 'Solutions Manual to Managerial Accounting' by Garrison, Ray H; Noreen, Eric W; Brewer Peter C (11th edition) - Chapter 10, 12 and 13. Please send it to my e-mail address at === Subject: Re: Solution Manual for Semiconductor Physics and Devices 3ed (Neamen) Sakhawat : > hi, please anyone send me the Solution Manual for Semiconductor Physics and Devices 3rd edition(Donald A. Neamen) > its urgent. exam this week. my email: bulbul88@gmail.com > Sakhawat. === Subject: The more big list of solutions manual (for all the siubjects) My List of Solutions Manual (Not exchange) contact me to : bergh123@yahoo.com If your solutions manual wanted ins't on this list, also can make the request. These are some only. 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Jagabanduhu Chakrabarty) Engineering Materials Vol. 1 : An Introduction to Properties, Applications and Design (3rd Ed., Michael Ashby & David R H Jones) Engineering Materials Vol. 2 : An Introduction to Microstructures, Processing and Design (3rd Ed., Michael Ashby & David R H Jones) Plastics: Microstructure and Engineering Applications (3rd Ed., Nigel Mills) Laser Processing of Engineering Materials : Principles, Procedure and Industrial Application (John Ion) Sintering : Densification, Grain Growth and Microstructure (Suk-Joong Kang) Biomaterials Science : An Introduction to Materials in Medicine (2nd Ed., Buddy D. Ratner, Allan Hoffman, Frederick Schoen & Jack Lemons) === Subject: Re: JSH: War of attrition > Archimedes said that with a level long enough and a place to stand he > could move the world because he could conceive of greatness on a scale > that most people cannot. > Where in the world did you get such an interpretation of this Archimedes quote? I'm quite certain that Archimedes was simply referring to the extraordinary law of the lever without any overtones of his sense of position in society or history. I can think of nothing in reliable biographies of Archimedes, and certainly not his own writings, that would suggest that he would use mathematics or natural philosophy as a metaphor like this. Archimedes is a very important person in human history (ironically), and should not be misrepresented like this. As penance, you should read Geometrical solutions derived from mechanics http://ia300213.us.archive.org/3/items/117719329/117719329.pdf === Subject: UNDERWEAR LINGERIE Lingerie is a term for women's fashion underwear. It derives from the French word 'lin' for linen. While the term in the French language applies to all underwears for either sex, in English it is applied only to those women's underwears designed to be visually appealing or erotic, typically incorporating materials such as nylon (nylon tricot), polyester, satin, lace and/or silk, and not applied to functional cotton underwears. Huge collection of beautiful lingerie, find your favorite lingerie in this lingerie gallery, many models of lingeries are shown here, from the vintage lingerie style to the latest one. http://lingeriegallery.blogspot.com/ === Subject: Re: UNDERWEAR LINGERIE > Lingerie is a term for women's fashion underwear. It derives from the > French word 'lin' for linen. While the term in the French language > applies to all underwears for either sex, in English it is applied > only to those women's underwears designed to be visually appealing or > erotic, typically incorporating materials such as nylon (nylon > tricot), polyester, satin, lace and/or silk, and not applied to > functional cotton underwears. Also it is almost invariably mispronounced. -- Corporate society looks after everything. All it asks of anyone, all it has ever asked of anyone, is that they do not interfere with management === Subject: need solutions i need the solutions to fluid mechanics- fundamentals and applications 1st ed. 2006 isbn: 0072472367 authors: cengel & cimbala email me with what yo do next === Subject: solutions manual i need the solutions manual for fluid mechanics- fundamentals and applications 1ys ed. 2006 isbn: 0072472367 authors: cengel and cimbala email me at wolfe8105@yahoo.com === Subject: Re: +1600 Solutions Manual to low cost hello, i need the solutions man. for fluid mechanics - fundamentals and applications 1st ed. 2006 isbn: 0072472367 authors: cengel and cimbala email me at wolfe8105@yahoo.com === Subject: Re: Complete solution manual in PDF format ....only $6.50usd(each) <4663CC3F.CCA9AEAD@antispamhotmail.co.uk.invalid> Just to let everyone know, you can get these for free on cramster.com On Jun 4, 3:24 am, Frederick Williams i only accept paypal,if you have credit card do sign up for new account at (www.paypal.com) > I have the solutions manual in electronic format for the following textbooks. They include solutions to the problems in the text. Payment is through Paypal for the amount of US $6.50 each. When sending payment ,please mention which solution manual you want and the email to send the file to, the file will be sent within 24 hours as the payment received.As soon as the payment received, a download link for the solution manual will be sent to your email address > Email me at cheapestsolut...@hotmail.com > ... hi there > i would like to have : Modern Control System 9th Edition by Drof, > Bishop > 10th edition will be very helpful , but as u have 9th , i can try > with it ..please let me now ur e-mail where i can sent 6.50 as u > said ....thanx > rahma28 Three points: Note that the op says Don't post replies here [...] Email me at > cheapestsolut...@hotmail.com. Snip most of the op's post to save bandwidth. Write proper English, i, u, thanx may be appropriate for mindless > teenagers texting one another but it is not appropriate for an educated > person. -- > Remove antispam and .invalid for e-mail address. > He that giveth to the poor lendeth to the Lord, and shall be repaid, > said Mrs Fairchild, hastily slipping a shilling into the poor woman's > hand. === Subject: Re: Proposition 1, about a number between twin prime numbers. Not always. Here is a counter example. 1234627 and 1234629 are twin primes. 1234628 has factors 2, 4, 257, 514, 1028, 1201, 2402, 4804, 308657, 617314 their sum 2 + 4+ 257+ 514+ 1028+ 1201+ 2402+ 4804+ 308657+ 617314 = 936,183 which is less than 1,234,628 therefore not abundant (deficient). === Subject: Re: Proposition 1, about a number between twin prime numbers. <9144726.1182372660604.JavaMail.jakarta@nitrogen.mathforum.org Not always. > Here is a counter example. > 1234627 and 1234629 > are twin primes. 1234628 has factors > 2, 4, 257, 514, 1028, 1201, 2402, 4804, 308657, 617314 their sum > 2 + 4+ 257+ 514+ 1028+ 1201+ 2402+ 4804+ 308657+ 617314 > = 936,183 which is less than 1,234,628 > therefore not abundant (deficient). much simpler counterexample: 3 and 5 are twin primes. the factors of 4 are 1, 2, 4. These sum to 7, and so 4 is deficient, not abundant. === Subject: Re: Proposition 1, about a number between twin prime numbers. On Wed, 20 Jun 2007 16:50:30 EDT, Prime Digits Here is a counter example. >1234627 and 1234629 >are twin primes. > 1234629 = (3^3)*11*4157 --Lynn === Subject: Re: Solution manual of textbooks in ebook format! Get it in hours! <25072386.1178970034064.JavaMail.jakarta@nitrogen.mathforum.org Hudreds and Hundreds of Solutions Manuals and Ebooks in all Subjects. All Solutions Manuals and Ebooks are in pdf format or Microsoft Word format. To get the Solutions Manualsor Ebooks, just email me with a price you want to get the solutions manual you want. > To see samples, just email me. The Solutions Manuals and Ebooks will be updated frequently. > If you can't find the solution manual listed below, don't give up, feel free to email me My email is edisonee(at)gmail.com please replace the (at) with @ Solutions Manuals NET 2.0 Interoperability Recipes by Bruce Bukovics > A Course in Game Theory by Martin J. 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Shankar > Investment Analysis and Portfolio Management by ... read more E === Subject: Re: Solution manual of textbooks in ebook format! Get it in hours! <12615223.1178094572750.JavaMail.jakarta@nitrogen.mathforum.org > For most of the books given above I too have solution manuals. If anyone wants them please message me. I will send them to you free of cost!!! No money required.!! buddy can u plz plz send me solns for microelectronics by millman and > grabel.i have an exam day after tomorrow.... i need the solutions for fluids mechanics- fundamentals and applications 1st ed. 2006 isbn:0072472367 authors: cengel and cimbala email me at : wolfe8105@yahoo.com === Subject: Re: Solution manual of textbooks in ebook format! Get it in hours! <12615223.1178094572750.JavaMail.jakarta@nitrogen.mathforum.org> hey i m really in need of solution manual of Elements of Electromagnetic.Matthew N.O Sadiku.Will u kindly can send me === Subject: Re: Solution manual of textbooks in ebook format! Get it in hours! <12615223.1178094572750.JavaMail.jakarta@nitrogen.mathforum.org> hey i m really in need of solution manual of Elements of Electromagnetic.Matthew N.O Sadiku.Will u kindly can send me === Subject: Re: Solution manual of textbooks in ebook format! Get it in hours! <25072386.1178970034064.JavaMail.jakarta@nitrogen.mathforum.org Hudreds and Hundreds of Solutions Manuals and Ebooks in all Subjects. All Solutions Manuals and Ebooks are in pdf format or Microsoft Word format. To get the Solutions Manualsor Ebooks, just email me with a price you want to get the solutions manual you want. > To see samples, just email me. The Solutions Manuals and Ebooks will be updated frequently. > If you can't find the solution manual listed below, don't give up, feel free to email me My email is edisonee(at)gmail.com please replace the (at) with @ Solutions Manuals NET 2.0 Interoperability Recipes by Bruce Bukovics > A Course in Game Theory by Martin J. 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Shankar > Investment Analysis and Portfolio Management by ... read more ? Hi! I need the solution manual for Computer Networking A top-down approach 3rd Ed. by James F. Kurose . I am an international student with no funding.Please let me know if u can offer any discount. I can pay 8$ for ur solutions manual as u have quoted before. Do you offer Vijaya. === Subject: help, i'm having a brain fart For the love of creation, I can't remember how the -1 appears in factoring y - y * cot27.2*tan23.9 = 17.5*tan23.9 to y = 17.5*tan23.9 / 1 - tan23.9*cot27.2 === Subject: Re: help, i'm having a brain fart : For the love of creation, I can't remember how the -1 appears in factoring y - y * cot27.2*tan23.9 = 17.5*tan23.9 y - y*const = y*1 - y*const = y*(1-const). > y = 17.5*tan23.9 / 1 - tan23.9*cot27.2 This is wrong. It should be y = 17.5*tan 23.9 / (1 - tan 23.9 cot 27.2) Some parentheses would be good around the function arguments, but they are *essential* around the denominator. Then I would divide top and bottom by tan 23.9 to get y = 17.5 / (cot 23.9 - cot 27.2) I don't immediately see a way to simplify further. The sine and cosine have sum-to-product formulas, but I don't know of any such for tangent and cotangent. -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com/ === Subject: Re: help, i'm having a brain fart : For the love of creation, I can't remember how the -1 appears in factoring y - y * cot27.2*tan23.9 = 17.5*tan23.9 y - y*const = y*1 - y*const = y*(1-const). y = 17.5*tan23.9 / 1 - tan23.9*cot27.2 This is wrong. It should be y = 17.5*tan 23.9 / (1 - tan 23.9 cot 27.2) Some parentheses would be good around the function arguments, but > they are *essential* around the denominator. Then I would divide top and bottom by tan 23.9 to get y = 17.5 / (cot 23.9 - cot 27.2) I don't immediately see a way to simplify further. The sine and > cosine have sum-to-product formulas, but I don't know of any such for > tangent and cotangent. > Rewrite cotangent as cosine/sine, then you can multiply numerator and denominator by sin 23.9 *sin 27.2, and then use the formula for sin (x-y) to get: y = 17.5 * sin 23.9 * sin 27.2 / sin 3.3 I don't know if you would consider this simpler or not (sine is simpler than cotangent, and there are only products of functions now, but you have more of them, and they are in both halves of the fraction.) === Subject: Re: help, i'm having a brain fart Doh!, mistake, it's the 1 - For the love of creation, I can't remember how the -1 appears in > factoring y - y * cot27.2*tan23.9 = 17.5*tan23.9 > to > y = 17.5*tan23.9 / 1 - tan23.9*cot27.2 === Subject: Re: help, i'm having a brain fart > Doh!, mistake, it's the 1 - > For the love of creation, I can't remember how the -1 appears in > factoring y - y * cot27.2*tan23.9 = 17.5*tan23.9 > to > y = 17.5*tan23.9 / 1 - tan23.9*cot27.2 You've got y - y*A = B, so y(1 - A) = B, so y = B/(1 -A). -- Paul Sperry Columbia, SC (USA) === Subject: Re: help, i'm having a brain fart Doh!, mistake, it's the 1 - > For the love of creation, I can't remember how the -1 appears in > factoring y - y * cot27.2*tan23.9 = 17.5*tan23.9 > to > y = 17.5*tan23.9 / 1 - tan23.9*cot27.2 You've got y - y*A = B, so > y(1 - A) = B, so > y = B/(1 -A). > -- Any society that would give up a little liberty to gain a little security deserves neither and will lose both - Benjamin Franklin === Subject: Trigonometry Can you simplify this expression in less than 10 steps. { cosA + (sin2A/secA)} * sec 2A * (1 - tan A) --------------------------------------------------------------------------- [CapitalEth]------- Tarun Ayitam FIITJEE Junior College India === Subject: Trigonometry2 Please give me a very short solution and impressive solution for this equation. 2sec ^2 A - cos ^2 A - tan A ( 1 + sin A + sec A ) = 0 ---------------------------------- Tarun Ayitam FIITJEE Junior College India === Subject: Re: Proposition 2, about perfect numbers and abundant numbers. > Proposition 2: If p is an odd perfect number, then, p + 2 is an abundant number. ACan you obtain a proof of that? > The proposition is almost certainly false, even if OPN's exist. There are many requirements on OPN's that are fully proved. These show that an OPN must have LOTS of small prime factors (and at least 8 distinct ones) and can NOT be squarefree. If n has such properties, n+2 will, on average NOT have very many prime factors and indeed, its smallest prime factor will not be very small. This makes it unlikely that n+2 is abundant. === Subject: Re: Proposition 2, about perfect numbers and abundant numbers. Not True. Counter Example, the 10th perfect number is 191561942608236107294793378084303638130997321548169216 and 191561942608236107294793378084303638130997321548169218 factors to 2, 3, 6, 63853980869412035764931126028101212710332440516056406, 95780971304118053647396689042151819065498660774084609 which sums to 159634952173530089412327815070253031775831101290141026 and is therefor deficient. === Subject: Re: Proposition 2, about perfect numbers and abundant numbers. <18298180.1182389134486.JavaMail.jakarta@nitrogen.mathforum.org Not True. > Counter Example, the 10th perfect number is > 191561942608236107294793378084303638130997321548169216 > and > 191561942608236107294793378084303638130997321548169218 > factors to > 2, 3, 6, 63853980869412035764931126028101212710332440516056406, 95780971304118053647396689042151819065498660774084609 > which sums to 159634952173530089412327815070253031775831101290141026 > and is therefor deficient. But that is not an odd number, which the OP specifies is a requirement, and as I mentioned above, may be an empty set making the claim vacuously true. === Subject: Re: Proposition 2, about perfect numbers and abundant numbers. On Jun 16, 1:50 pm, Carlos Alberto Carcagno First show that there does not exist an odd perfect number (still an open question, I believe), and then the proposition is vacuously true. === Subject: Re: cross-ratios of the base points and base lines of two conics Originator: ilya@powdermilk.math.berkeley.edu [A complimentary Cc of this posting was sent to David Madore > Let C and D be two plane conics (in sufficiently general position), > p1, p2, p3, p4 the four intersection points of C and D, and l1, l2, > l3, l4 the four common tangents (i.e., intersections of the dual > concics, C* and D*). Is the following statement true? After possibly > reordering, the cross-ratio of p1, p2, p3, p4 on C is equal to the > cross-ratio of l1, l2, l3, l4 on D* (which is the same as the > cross-ratio of the points of tangency of l1, l2, l3, l4 on D). a) It is enough to show this on a Zariski open subset of the moduli space; a') It is enough to show this on a subset the PSL(3)-orbit of which is Zariski open. Actually, we use a following modification of (a): b) On a Zariski open subset of the moduli space, the pair (C,D) is projectively equivalent to the pair (D*, C*). Obviously, (b) implies your statement. Moreover, one change (b) to the primed variant, so it enough to consider pairs C,D of the form A x^2 + B y^2 = 1 B x^2 + A x^2 = 1 - and for these pairs the statement is obvious. Hope this helps, Ilya === Subject: Re: cross-ratios of the base points and base lines of two conics Ilya Zakharevich in litteris scripsit: > the pair (C,D) is > projectively equivalent to the pair (D*, C*). Indeed, this is a very nice way of stating the problem! And I agree that it then becomes pretty obvious (I would tend to state it in terms the enlightenment. Geometrically this is still a bit unsatisfactory, because we would want to construct an explicit projective map taking a point of C to a tangent to D and vice versa. But I guess this involves choosing one of four possible mappings of the intersection points of C and D to the common tangents to C and D so it can't be canonical. Or one would want to map a conic with the four common intersection points with C and D to one with the four common tangents - or something. -- David A. Madore (david.madore@ens.fr, http://www.dma.ens.fr/~madore/ ) === Subject: Re: Graph Theory > Are there any algorithms to find the minumim number of vertices in a > graph so that any other vertices not in that set are at least adjacent > to one of the vertices in the chosen set? This is called a dominating set. Computing dominating sets is NP-complete. Achim -- ________________________________________________________________________ | _____/ | Achim Blumensath O/ ___/ | TU Darmstadt =o= / | www.mathematik.tu-darmstadt.de/~blumensath / o----| _______________________________________________________________________| === Subject: Re: Graph Theory |Are there any algorithms to find the minumim number of vertices in a |graph so that any other vertices not in that set are at least adjacent |to one of the vertices in the chosen set? Yes, but I've never heard of a very good one. It's an NP-complete problem. See: http://en.wikipedia.org/wiki/Vertex_cover_problem Keith Ramsay === Subject: Re: Graph Theory What you are seeking is the domination number of a graph. For some references see, for example: http://en.wikipedia.org/wiki/Domination_number Various algorithms can be found in the books listed there === Subject: Re: Graph Theory references see, for example: > http://en.wikipedia.org/wiki/Domination_number Various algorithms can be found in the books listed there Does anyone know where I can find references to Minimum domination set algorithms? Paulo === Subject: Re: Graph Theory Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) >Does anyone know where I can find references to Minimum domination >set algorithms? Click on the link (in Wikipedia) for the Compendium of NP Optimization Problems. http://www.nada.kth.se/~viggo/wwwcompendium/node11.html -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: Essential singularity. I'm pretty sure I'm right about this, but can someone just confirm for me please: The function f(z) = 1/(z-1/2) + 1/(2(z-1/3)) + 1/(4(z-1/4)) + 1/ (8(z-1/5)) + ... has an essential singularity at the origin, right? And can someone also please confirm, that for a function with a natural boundary, all the points in the boundary are essential singularities of the function? -- Necessary evils are seldom necessary, but always evil! === Subject: Re: Essential singularity. I'm pretty sure I'm right about this, but can someone just confirm for > me please: The function f(z) = 1/(z-1/2) + 1/(2(z-1/3)) + 1/(4(z-1/4)) + 1/ > (8(z-1/5)) + ... has an essential singularity at the origin, right? Wrong. It's the limit of a sequence of poles. > And can someone also please confirm, that for a function with > a natural boundary, all the points in the boundary are > essential singularities of the function? None of them are. By definition, an essential singularity is an isolated singularity: the function must be analytic in a deleted neighbourhood of the point. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 === Subject: Re: Essential singularity. > I'm pretty sure I'm right about this, but can someone just confirm for > me please: The function f(z) = 1/(z-1/2) + 1/(2(z-1/3)) + 1/(4(z-1/4)) + 1/ > (8(z-1/5)) + ... has an essential singularity at the origin, right? > I looked up essential singularity at Mathworld, Wikipedia, PlanetMath, and Springer Encyclopaedia of Mathematics. Since your function f is not holomorphic in a punctured neighborhood of 0, I guess the answer is: no, 0 is not an essential singularity of f... -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Glasnik Matematicki - Mardesic issue New issue (June 2007) of the journal Glasnik Matematicki http://web.math.hr/glasnik/vol_42/vol42no1.html is dedicated to Professor Sibe Mardesic on the occasion of his 80th birthday http://web.math.hr/glasnik/vol_42/mardesic.html . This issue carries the following papers: A. Dujella, On Mordell-Weil groups of elliptic curves induced by Diophantine triples http://web.math.hr/glasnik/vol_42/no1_01.html V. Matijevic, Finite-sheeted covering maps over Klein bottle weakly solenoidal spaces http://web.math.hr/glasnik/vol_42/no1_02.html B. Raines and S. Stimac, Structure of inverse limit spaces of tent maps with nonrecurent critical points http://web.math.hr/glasnik/vol_42/no1_03.html A. Bogdewicz, R. J. MacG. Dawson and M. Moszynska, Chebyshev sets in hyperspaces over a Minkowski space http://web.math.hr/glasnik/vol_42/no1_04.html N. Antonyan, S. A. Antonyan and A. Soria-P.8erez, Homotopy characterization of G-ANR's http://web.math.hr/glasnik/vol_42/no1_05.html I. Ivansic and L. R. Rubin, Local characterization of absolute co-extensors http://web.math.hr/glasnik/vol_42/no1_06.html K. Eda, U. Karimov and D. Repovs, On the fundamental group of R^3 modulo the Chase-Chaberlin continuum http://web.math.hr/glasnik/vol_42/no1_07.html I. Ivansic and U. Milutinovic, Closed embeddings into Lipscomb's universal space http://web.math.hr/glasnik/vol_42/no1_08.html J. Segal and S. Spiez, A cohomological characteritazion of shape dimension for some class of spaces http://web.math.hr/glasnik/vol_42/no1_09.html J. J. Sanchez-Gabites and J. M. R. Sanjurjo, Shape properties of the boundary of attractors http://web.math.hr/glasnik/vol_42/no1_10.html P. Pavesic, A spectral sequence for the group of self-maps which induce identity automorphisms of homology groups http://web.math.hr/glasnik/vol_42/no1_11.html N. Koceic Bilan and N. Uglesic, The coarse shape http://web.math.hr/glasnik/vol_42/no1_12.html S. Nowak, The stable shape of compact spaces with countable cohomological groups http://web.math.hr/glasnik/vol_42/no1_13.html N. Uglesic and B. Cervar, The S_n-equivalence of compacta http://web.math.hr/glasnik/vol_42/no1_14.html M. Bestvina and K. Fujiwara, Quasi-homomorphisms on mapping class groups http://web.math.hr/glasnik/vol_42/no1_15.html S. Jabuka, Heegaard Floer homology and knot concordance: A survey of recent advances http://web.math.hr/glasnik/vol_42/no1_16.html Andrej Dujella managing editor of Glasnik Matematicki http://web.math.hr/glasnik/ === Subject: Re: 1-exp(1)+exp(exp(1))-exp(exp(exp(1))) + - ...=? Oscar - putting some aspects of your previous posts together. The best presentation of my method is to say that it provides for S(s,x) = x - s^x + s^s^x - s^s^s^x + ... - (*1a) here for x=1 S(s):= S(s,1) = 1 - s + s^s - s^s^s + ... - (*1b) a polynomial representation in terms of coefficients as_k depending on s S(s,x) = as_0 + as_1 x + as_2 x^2 + as_3 x^3 + ... (*2a) S(s,1) = as_0 + as_1 + as_2 + as_3 + ... (*2b) S(s,0) = as_0 (*2c) For the documentation of the behaviour of S(s) in my method, I make use of the following identity: It is according to (*1a) and (*1b) S(s,1) = 1 - s^1 + s^s^1 - s^s^s^1 + ... - S(s,0) = 0 - 1 + s^1 - s^s^1 + s^s^s^1 + ... - from where in the limit it should be S(s,1) = - S(s,0) (*3) By (*2b) and (*2c) it is S(s,1) = as_0 + as_1 + as_2 + as_3 + ... S(s,0) = as_0 and since by (*3) S(s,1) = - S(s,0) we have, that S(s,1) is completely explained by the first of the coefficients as_0 and thus S(s) = - as_0 Now as_0 is itself the result of a vector-product, and to check its behaviour I document its individual additive components as_0 = bs_0 + bs_1 + bs_2 + ... so that the best simplificated result of my method is then (assuming x=1) S(s) = - ( bs_0 + bs_1 + bs_2 + bs_3 + ... ) (*4) --------------------------------------------------------------------- > Part 1: The approach We have seen what the sum (both Euler's and yours) is at b = sqrt(2) = > 1.414+. Examine the Euler sum for b = 1.43, 1.44, 1.444. You can also > try your method if you wish for these bases, although the result for b = > sqrt(2) seems to indicate that you'll just get back the Euler sum. Let T(s) denote the infinite powertower s^s^s^s^s... . For s providing the convergent cases T(s) I crosschecked my method (*4) for computing S(s) by usual Eulersummation of the terms (*1b), where I got identity in terms of float-arithmetic approximation. That means all s in e^-e +eps < s < e^(1/e), where eps ~ 2^-12; this includes also s=sqrt(2). Part 2: Over the edge If Part 1 appears satisfactory, examine the sum by your method (Euler's > will no longer work) for b = 1.45, 1.46, 1.47. Compare these results > with the earlier ones for b = 1.43-1.444, to see if there is reasonably > smooth behavior. If yes, you have an apparently suitable analytic > continuation that can be applied for large values of b such as e. For the coefficients bs_k in (*4) I get nicely decreasing absolute values of alternating signs, where however the alternating behaviour is not strict. a small table for various s param ! coefficients sum(- bs_k) s ! bs_0 bs_1 bs_2 bs_3 bs_4 ... S(s) -------+-------------------------------------------------------------------- - 1 ! 0 -0.5 0 0 0 ... 1/2 1.0625 ! 0 -0.47 0.00076 1e-6 2e-8 ... 0.4706 1.4375 ! 0 -0.367 0.009 -6e-6 1e-5 .(decreasing) 0.3576 1.5 ! 0 -0.356 0.010 -1e-4 5e-5 .(decreasing) 0.3461 3 ! 0 -0.238 -0.002 -2e-4 6e-4 .(decreasing) 0.2368 7 ! 0 -0.170 -0.017 4e-3 2e-3 .(decreasing) 0.1795 e^2 ! 0 -0.167 -0.018 4e-3 2e-3 0.1769 e^3 ! 0 -0.125 -0.023 2e-4 3e-3 0.1421 where for higher s the dominant term is further decreasing in absolute value and with a vague floating the subsequent terms are still decreasing, while the rate of decrease seems to diminuish. But this may be due to the finite dimension of my matrices and I'll check this later. For s = exp(k) I seem to get nice approximations to rationals for bs_1, saying bs_1 = -1/2 * 1/(k+1) which explains then the method-internal singularity at s = exp(-1). With decreasing s I get less bs_k which are negative, so the alternating character disppears and at s = e^(-e) I get a series with only positive terms, which also increase, and thus cannot be summed to a finite value. param ! coefficients sum(- bs_k) s ! bs_0 bs_1 bs_2 bs_3 bs_4 ... S(s) -------+-------------------------------------------------------------------- - e^(-e) ! 0 0.291 1.332 2.369 3.210 -inf- > The singularity I was referring to has to do with the terms of the > series that suddenly go from bounded to increasing extremely fast. > Therefore there is a catastrophic breakdown of the Euler summation. But it turns out that the iterated Shanks extrapolation continues to > converge, and that its results indicate continuous behavior across the > Euler-summation singularity. Following are sums I obtained using > Microsoft Excel: Base Sum > 1.43 0.359117 > 1.44 0.357152 > -------Euler-summation Singularity-------- > 1.45 0.355226 > 1.46 0.353346* > 1.47 0.351477* *Slight uncertainty since I could get only a few partial sums before the > terms overflowed. The implication is that sums are definable across the Euler-summation > sigularity; that is, the singularity is removable. But for bases > substantially larger than e^(1/e) in my notation, the Shanks mthod > becomes computationally unfeasible and we would have to rely on your > method. > I couldn't reproduce the behave of the Euler-summation. The terms to be summed are for the method (*1b), s= sqrt(2) 1 -1.4142136 2.0000000 -2.8284271 4.0000000 -5.6568542 8.0000000 -11.313708 If I apply my implementation of Eulersummation I get the following smooth sequence of partial sums: 0.50000000 0.39644661 0.41789322 0.41345148 0.41437139 0.41418087 which stabilizes at about the 10'th term in the last displayed digit. Here I don't know, what's going on compared to your report? Gottfried --- Gottfried Helms, Kassel === Subject: Re: 1-exp(1)+exp(exp(1))-exp(exp(exp(1))) + - ...=? / another correction Am 29.06.2007 15:46 schrieb Gottfried Helms: This was crap. Wrong parameter setting: I couldn't reproduce the behave of the Euler-summation. > The terms to be summed are for the method (*1b), s= sqrt(2) 1 -1.4142136 2.0000000 -2.8284271 4.0000000 -5.6568542 8.0000000 -11.313708 If I apply my implementation of Eulersummation I get the following > smooth sequence of partial sums: 0.50000000 0.39644661 0.41789322 0.41345148 0.41437139 0.41418087 which stabilizes at about the 10'th term in the last displayed digit. > It must be: --------------------------------------------------- The terms to be summed are for the method (*1b), s= sqrt(2) 1, -1.4142136, 1.6325269, -1.7608396, 1.8409109, -1.8927127, 1.9269997, ... If I apply my implementation of Eulersummation I get the following smooth sequence of partial sums: 0.50000000 0.39644661 0.37195908 0.36534037 0.36333599 0.36267423 which stabilizes at about the 20'th term in the last displayed digit to 0.36230155 ----------------------------------------------------- Gottfried --- Gottfried Helms, Kassel