mm-1433 === Subject: Re: A curious parallel Alright, since we have you types here for now, even though it might *still* be off topic 'cause it's maybe a physics thing, Some of us maintain that when you turn a pair of skis, your body, or the appropriate majority portion there of, must be on the inside of the turn/skis as they turn. Some like to say center of mass AOT body. The reason being, inertia will tend to make the skier continue to go straight, as such, if the body was on the outside of the skis as the skier tried to turn, the skier would fall to the outside. It's a principle called crossover. Cross your body over the skis, or cross the skis under the body, anyway, same principle. Oui? Non? === Subject: Re: Good morning or good evening depending upon your location. I want to ask you the most important question of your life. Your joy or sorrow for all eternity depends upon your answer. The question is: Are you saved? It is not a question of how good you ar Does my 401(k) count as savings? --Keith This is the most important question of your life. The question is: Are you saved? It is not a question of how good you are, nor if you are a church member, but are you saved? Are you sure you will go to Heaven when you die? The reason some people don't know for sure if they are going to Heaven when they die is because they just don't know. The good news is that you can know for sure that you are going to Heaven. The Holy Bible describes Heaven as a beautiful place with no death, sorrow, sickness or pain. God tells us in the Holy Bible how simple it is to be saved so that we can live forever with Him in Heaven. For if you confess with your mouth Jesus is Lord and believe in your heart that God raised Him from the dead, you WILL BE SAVED. (Romans 10:9) Over 2000 years ago God came from Heaven to earth in the person of Jesus Christ to shed His blood and die on a cross to pay our sin debt in full. Jesus Christ was born in Israel supernaturally to a virgin Jewish woman named Mary and lived a sinless life for thirty-three years. At the age of thirty-three Jesus was scourged and had a crown of thorns pressed onto His head then Jesus was crucified. Three days after Jesus died on a cross and was placed in a grave Jesus rose from the dead as Jesus said would happen before Jesus died. If someone tells you that they are going to die and then three days later come back to life and it actually happens then this person must be the real deal. Jesus Christ is the only person that ever lived a perfect sinless life. This is why Jesus is able to cover our sins(misdeeds) with His own blood because Jesus is sinless. The Holy Bible says, In Him(Jesus) we have redemption through His blood, the forgiveness of sins... (Ephesians 1:7) If you would like God to forgive you of your past, present and future sins just ask Jesus Christ to be your Lord and Saviour. It doesn't matter how old you are or how many bad things that you have done in your life including lying and stealing all the way up to murder. Just pray the prayer below with your mouth and mean it from your heart and God will hear you and save you. have a home in Heaven with You when I die. I agree with You that I am a sinner. I believe that You love me and want to save me. I believe that You bled and died on the cross to pay the penalty for my sins and that You rose from the dead. Please forgive my sins and come into my heart and be my Lord and Saviour. me through Your merciful grace. Amen. Welcome to the family of God if you just allowed God to save you. Now you are a real Christian and you can know for sure that you will live in Heaven forever when this life comes to an end. As a child of God we are to avoid sin(wrongdoing), but if you do sin the Holy Bible says, My dear children, I write this to you so that you will not sin. But if anybody does sin, we have one who speaks to the Father in our defense Jesus Christ, the Righteous One. Those of you that have not yet decided to place your trust in the Lord Jesus Christ may never get another chance to do so because you do not know when you will die. Jesus said, I am the way, the truth and the life: no one can come to the Father(God)(in Heaven), but by me. (John 14:6) This means that if you die without trusting in Jesus Christ as your Lord and Saviour you will die in your sins and be forever separated from the love of God in a place called Hell. The Holy Bible descibes Hell as a place of eternal torment, suffering, pain and agony for all those who have rejected Jesus Christ. The good news is that you can avoid Hell by allowing Jesus Christ to save you today. Only then will you have true peace in your life knowing that no matter what happens you are on your way to Heaven. Praise the Lord! Servant of the Lord Jesus Christ Ronald L. Grossi *Show this to your family and friends so they can know that they have a choice ...87.bcÁ`Á.bc.87...[AAcute ].bcÁ`Á.bc.87.?..87`[DownExc lamation].bc.87`Á.bc.87.....87.bc[Down Exclamation]`Á.bc.87. Got Questions? http://www.gotquestions.org/archive.html Other Languages http://www.godssimpleplan.org/gsps.html Free Movie: To Hell and Back http://www.tbn.org/index.php/8/1.html Animation http://www.gieson.com/Library/projects/animations/walk/index.html The Passion Of The Christ http://www.thepassionofthechrist.com Beware Of Cults http://www.carm.org/cults/cultlist.htm About Hell http://www.equip.org/free/DH198.htm Is Jesus God? http://www.powertochange.com/questions/qna2.html Free Online Bible http://www.biblegateway.com ...87.bcÁ`Á.bc.87...[AAcute ].bcÁ`Á.bc.87.?..87`[DownExc lamation].bc.87`Á.bc.87.....87.bc[Down Exclamation]`Á.bc.87. *This message may get deleted so you may want to print a copy. *Just press the [Ctrl][P] keys on your keyboard to print this page. === Subject: Re: JSH: SFT and integers > The problem I face is that posters here engage in pseudo-math > arguments, so I have to slowly figure out what is hard for them to > confuse others over. > Here I think it might help to emphasize that the SFT solves > sqrt(u^2 - 4A^2(A^2 - B^2) v^2) > where all are non-zero integers, and > u/v = (z - 2A^2)^2. z = x(x +/- sqrt((x - 2B^2)^2 + 4B^2 (A^2 - B^2)))/(2x - 2A^2) and x is given by x = +/- (g_1 - g_2) + 2B^2 where g_1 g_2 = B^2(A^2 - B^2). Where g_1 and g_2 are rationals but it's easier to make substitutions first and then move fully to integers. Picking the positive square root for x, gives x = g_1 - g_2 + 2B^2 so I have z = ((g_1 - g_2) + 2B^2)(g_1 - g_2 + 2B^2 +/-(g_1 + g_2))/(2(g_1 - g_2 + 2B^2 - A^2)) and taking the positive again z = ((g_1 - g_2) + 2B^2)(g_1 + B^2))/(g_1 - g_2 + 2B^2 - A^2) and now I can move to integers by using g_1 = b_1 (d_2)/d_1 g_2 = b_2 (d_1)/d_2 where b_1 b_2 = B^2(A^2 - B^2) but are integers, and d_1 and d_2 are non-zero integers as well. Notice that g_1 g_2 = b_1 b_2. Then I have z = (((b_1 d_2)/d_1 - (b_2 d_1)/d_2) + 2B^2)((b_1 d_2)/d_1 + B^2))/((b_1 d_2)/d_1 - (b_2 d_1)/d_2 + 2B^2 - A^2) which is z = ((b_1 d_2^2 - b_2 d_1^2 + 2B^2 d_1 d_2)(b_1 d_2 + B^2 d_1))/d_1(b_1 d_2^2 - b_2 d_1^2 + (2B^2 - A^2)d_1 d_2) and that's interesting, as it looks like d_1 will have to be a factor of b_1 d_2 if there is any possibility of z being an integer. So, let d_1 = 1, which should simplify things a bit: z = ((b_1 d_2^2 - b_2 + 2B^2 d_2)(b_1 d_2 + B^2))/(b_1 d_2^2 - b_2 + (2B^2 - A^2)d_2) and taking a pause it looks like I'll have a cubic in the numerator with a quadratic in the denominator, and it's just a matter of dividing through. That will leave some remainder which will have the conditions necessary for d_2 so that z will be an integer. James Harris === Subject: Re: JSH: SFT and integers >> The problem I face is that posters here engage in pseudo-math >> arguments, so I have to slowly figure out what is hard for them to >> confuse others over. >> Here I think it might help to emphasize that the SFT solves >> sqrt(u^2 - 4A^2(A^2 - B^2) v^2) >> where all are non-zero integers, and >> u/v = (z - 2A^2)^2. >z = x(x +/- sqrt((x - 2B^2)^2 + 4B^2 (A^2 - B^2)))/(2x - 2A^2) >and x is given by >x = +/- (g_1 - g_2) + 2B^2 >where >g_1 g_2 = B^2(A^2 - B^2). >Where g_1 and g_2 are rationals but it's easier to make substitutions >first and then move fully to integers. >Picking the positive square root for x, gives >x = g_1 - g_2 + 2B^2 >so I have >z = ((g_1 - g_2) + 2B^2)(g_1 - g_2 + 2B^2 +/-(g_1 + g_2))/(2(g_1 - g_2 >+ 2B^2 - A^2)) >and taking the positive again >z = ((g_1 - g_2) + 2B^2)(g_1 + B^2))/(g_1 - g_2 + 2B^2 - A^2) >and now I can move to integers by using >g_1 = b_1 (d_2)/d_1 >g_2 = b_2 (d_1)/d_2 >where >b_1 b_2 = B^2(A^2 - B^2) >but are integers, and d_1 and d_2 are non-zero integers as well. >Notice that g_1 g_2 = b_1 b_2. >Then I have >z = (((b_1 d_2)/d_1 - (b_2 d_1)/d_2) + 2B^2)((b_1 d_2)/d_1 + >B^2))/((b_1 d_2)/d_1 - (b_2 d_1)/d_2 + 2B^2 - A^2) >which is >z = ((b_1 d_2^2 - b_2 d_1^2 + 2B^2 d_1 d_2)(b_1 d_2 + B^2 d_1))/d_1(b_1 >d_2^2 - b_2 d_1^2 + (2B^2 - A^2)d_1 d_2) >and that's interesting, as it looks like d_1 will have to be a factor >of b_1 d_2 if there is any possibility of z being an integer. >So, let d_1 = 1, which should simplify things a bit: >z = ((b_1 d_2^2 - b_2 + 2B^2 d_2)(b_1 d_2 + B^2))/(b_1 d_2^2 - b_2 + >(2B^2 - A^2)d_2) >and taking a pause it looks like I'll have a cubic in the numerator >with a quadratic in the denominator, and it's just a matter of dividing >through. >That will leave some remainder which will have the conditions necessary >for d_2 so that z will be an integer. Great! So now you are done. Finished. No need to post anything more on this subject. Congratulations. J.9frgen >James Harris === Subject: Re: JSH: SFT and integers > [...] Your last five posts to this thread have all been the same. Have you run out of ideas already? --- Christopher Heckman === Subject: Re: JSH: SFT and integers > [...] Your last five posts to this thread have been the same. Have you run out of ideas already? --- Christopher Heckman === Subject: Re: JSH: SFT and integers > The problem I face is that posters here engage in pseudo-math > arguments, so I have to slowly figure out what is hard for them to > confuse others over. > Here I think it might help to emphasize that the SFT solves > sqrt(u^2 - 4A^2(A^2 - B^2) v^2) > where all are non-zero integers, and > u/v = (z - 2A^2)^2. z = x(x +/- sqrt((x - 2B^2)^2 + 4B^2 (A^2 - B^2)))/(2x - 2A^2) and x is given by x = +/- (g_1 - g_2) + 2B^2 where g_1 g_2 = B^2(A^2 - B^2). Where g_1 and g_2 are rationals but it's easier to make substitutions first and then move fully to integers. Picking the positive square root for x, gives x = g_1 - g_2 + 2B^2 so I have z = ((g_1 - g_2) + 2B^2)(g_1 - g_2 + 2B^2 +/-(g_1 + g_2))/(2(g_1 - g_2 + 2B^2 - A^2)) and taking the positive again z = ((g_1 - g_2) + 2B^2)(g_1 + B^2))/(g_1 - g_2 + 2B^2 - A^2) and now I can move to integers by using g_1 = b_1 (d_2)/d_1 g_2 = b_2 (d_1)/d_2 where b_1 b_2 = B^2(A^2 - B^2) but are integers, and d_1 and d_2 are non-zero integers as well. Notice that g_1 g_2 = b_1 b_2. Then I have z = (((b_1 d_2)/d_1 - (b_2 d_1)/d_2) + 2B^2)((b_1 d_2)/d_1 + B^2))/((b_1 d_2)/d_1 - (b_2 d_1)/d_2 + 2B^2 - A^2) which is z = ((b_1 d_2^2 - b_2 d_1^2 + 2B^2 d_1 d_2)(b_1 d_2 + B^2 d_1))/d_1(b_1 d_2^2 - b_2 d_1^2 + (2B^2 - A^2)d_1 d_2) and that's interesting, as it looks like d_1 will have to be a factor of b_1 d_2 if there is any possibility of z being an integer. So, let d_1 = 1, which should simplify things a bit: z = ((b_1 d_2^2 - b_2 + 2B^2 d_2)(b_1 d_2 + B^2))/(b_1 d_2^2 - b_2 + (2B^2 - A^2)d_2) and taking a pause it looks like I'll have a cubic in the numerator with a quadratic in the denominator, and it's just a matter of dividing through. That will leave some remainder which will have the conditions necessary for d_2 so that z will be an integer. James Harris === Subject: Re: JSH: SFT and integers > The problem I face is that posters here engage in pseudo-math > arguments, so I have to slowly figure out what is hard for them to > confuse others over. > Here I think it might help to emphasize that the SFT solves > sqrt(u^2 - 4A^2(A^2 - B^2) v^2) > where all are non-zero integers, and > u/v = (z - 2A^2)^2. z = x(x +/- sqrt((x - 2B^2)^2 + 4B^2 (A^2 - B^2)))/(2x - 2A^2) and x is given by x = +/- (g_1 - g_2) + 2B^2 where g_1 g_2 = B^2(A^2 - B^2). Where g_1 and g_2 are rationals but it's easier to make substitutions first and then move fully to integers. Picking the positive square root for x, gives x = g_1 - g_2 + 2B^2 so I have z = ((g_1 - g_2) + 2B^2)(g_1 - g_2 + 2B^2 +/-(g_1 + g_2))/(2(g_1 - g_2 + 2B^2 - A^2)) and taking the positive again z = ((g_1 - g_2) + 2B^2)(g_1 + B^2))/(g_1 - g_2 + 2B^2 - A^2) and now I can move to integers by using g_1 = b_1 (d_2)/d_1 g_2 = b_2 (d_1)/d_2 where b_1 b_2 = B^2(A^2 - B^2) but are integers, and d_1 and d_2 are non-zero integers as well. Notice that g_1 g_2 = b_1 b_2. Then I have z = (((b_1 d_2)/d_1 - (b_2 d_1)/d_2) + 2B^2)((b_1 d_2)/d_1 + B^2))/((b_1 d_2)/d_1 - (b_2 d_1)/d_2 + 2B^2 - A^2) which is z = ((b_1 d_2^2 - b_2 d_1^2 + 2B^2 d_1 d_2)(b_1 d_2 + B^2 d_1))/d_1(b_1 d_2^2 - b_2 d_1^2 + (2B^2 - A^2)d_1 d_2) and that's interesting, as it looks like d_1 will have to be a factor of b_1 d_2 if there is any possibility of z being an integer. So, let d_1 = 1, which should simplify things a bit: z = ((b_1 d_2^2 - b_2 + 2B^2 d_2)(b_1 d_2 + B^2))/(b_1 d_2^2 - b_2 + (2B^2 - A^2)d_2) and taking a pause it looks like I'll have a cubic in the numerator with a quadratic in the denominator, and it's just a matter of dividing through. That will leave some remainder which will have the conditions necessary for d_2 so that z will be an integer. James Harris === Subject: Re: JSH: SFT and integers > The problem I face is that posters here engage in pseudo-math > arguments, so I have to slowly figure out what is hard for them to > confuse others over. > Here I think it might help to emphasize that the SFT solves > sqrt(u^2 - 4A^2(A^2 - B^2) v^2) > where all are non-zero integers, and > u/v = (z - 2A^2)^2. z = x(x +/- sqrt((x - 2B^2)^2 + 4B^2 (A^2 - B^2)))/(2x - 2A^2) and x is given by x = +/- (g_1 - g_2) + 2B^2 where g_1 g_2 = B^2(A^2 - B^2). Where g_1 and g_2 are rationals but it's easier to make substitutions first and then move fully to integers. Picking the positive square root for x, gives x = g_1 - g_2 + 2B^2 so I have z = ((g_1 - g_2) + 2B^2)(g_1 - g_2 + 2B^2 +/-(g_1 + g_2))/(2(g_1 - g_2 + 2B^2 - A^2)) and taking the positive again z = ((g_1 - g_2) + 2B^2)(g_1 + B^2))/(g_1 - g_2 + 2B^2 - A^2) and now I can move to integers by using g_1 = b_1 (d_2)/d_1 g_2 = b_2 (d_1)/d_2 where b_1 b_2 = B^2(A^2 - B^2) but are integers, and d_1 and d_2 are non-zero integers as well. Notice that g_1 g_2 = b_1 b_2. Then I have z = (((b_1 d_2)/d_1 - (b_2 d_1)/d_2) + 2B^2)((b_1 d_2)/d_1 + B^2))/((b_1 d_2)/d_1 - (b_2 d_1)/d_2 + 2B^2 - A^2) which is z = ((b_1 d_2^2 - b_2 d_1^2 + 2B^2 d_1 d_2)(b_1 d_2 + B^2 d_1))/d_1(b_1 d_2^2 - b_2 d_1^2 + (2B^2 - A^2)d_1 d_2) and that's interesting, as it looks like d_1 will have to be a factor of b_1 d_2 if there is any possibility of z being an integer. So, let d_1 = 1, which should simplify things a bit: z = ((b_1 d_2^2 - b_2 + 2B^2 d_2)(b_1 d_2 + B^2))/(b_1 d_2^2 - b_2 + (2B^2 - A^2)d_2) and taking a pause it looks like I'll have a cubic in the numerator with a quadratic in the denominator, and it's just a matter of dividing through. That will leave some remainder which will have the conditions necessary for d_2 so that z will be an integer. James Harris === Subject: Continuity Let f(x) = sum_{k=1}^n (Cos[9^n*Pi*x]/2^n) prove f(x) is continuous but not differentiable everywhere. === Subject: Re: Continuity >Let f(x) = sum_{k=1}^n (Cos[9^n*Pi*x]/2^n) >prove f(x) is continuous but not differentiable everywhere. Oops. Didn't notice the typo until I saw William's reply. out, your definition is just f(x) = n*Cos[9^n*Pi*x]/2^n, which is surely not what you meant. You meant to say that f(x) = sum_{n=1}^infinity (Cos[9^n*Pi*x]/2^n). To put that another way, f = lim f_n, where f_n(x) = sum_{k=1}^n (Cos[9^k*Pi*x]/2^k) . ************************ David C. Ullrich === Subject: Re: Continuity I mean (a) f(x) is continuous everywhere (b) f(x) not differentiable everywhere. === Subject: Re: Continuity >Let f(x) = sum_{k=1}^n (Cos[9^n*Pi*x]/2^n) >prove f(x) is continuous If this is not extremely easy for you you don't have a chance with the second part. >but not differentiable everywhere. Here do you mean [i] not [differentiable everywhere], ie non-differentiable at at least one point, or [ii] [not differentiable] everywhere, ie differentiable at _no_ point? Both are true, but [i] is considerably easier. ************************ David C. Ullrich === Subject: Re: Continuity > Let f(x) = sum_{k=1}^n (Cos[9^n*Pi*x]/2^n) > prove f(x) is continuous but not differentiable everywhere. f(x) = sum_{k=1}^n (Cos[9^n*Pi*x]/2^n) = n(Cos[9^n*Pi*x]/2^n) f'(x) = -n.pi.(9/2)^n sin[9^n*Pi*x] === Subject: Re: proofs question >Wow it was staring at me all the time. Since the gcd of c and n is a >John You can also start by letting g = gcd(c, n). Then g divides both c and n. When g > 1, try to construct a and b such that [ac]_n = [bc]_n but [a]_n <> [b]_n. there exists a p -- If same-sex marriages are so bad, why do most college dormitories assign same-sex roommates? pmontgom@cwi.nl Microsoft Research and CWI Home: Bellevue, WA === Subject: Re: proofs question > I'm having problem on a proofs question for homework. If anyone could > give me some hints it would be appreciated. Here is the question. Let n > be a positive integer greater than or equal to 2 and let c be an > integer. Let f be the function from Z sub n to Z sub n defined by > f([a]) =[ca] for all [a] in Zn. Show that f is one to one > if and only if gcd (c,n)=1. > I can prove it going < way but I can't figure out how to prove it going > way. I've tried contrapositive and contradiction. This is what I have > so far. > If I assume f is one-to-one, I get these assumptions. > Let [a]=[b]. Then [ca]=[cb]. Actually, this follows whether f is one-to-one or not. Better would be: Assume [ca]=[cb]. Then [a]=[b] since f is 1-1. But, as someone else already mentioned, where are you getting these assumptions: what is b? what is a? It appears that your approach is similar to the way one does epsilon-delta proofs in calculus: assume epsilon given, take some unspecified delta, work out the conditions, and finally express delta in terms of epsilon. That is, you are working out the proof backwards. > So a congruent b (mod n) and ca congruent > cb (mod n). Therefore a-b=ns and ca-cb=nr for integers r and s. Let > d=gcd(c,n), then cx+cy=d for integers x and y. I assume you meant cx + ny = d. >I'm not really sure > where to go from here. If you could give me some help and not the > answer that would be appreciated. > John You have ca-cb=nr. Then, (c/d)a - (c/d)b = (n/d)r. Since gcd of c/d and n/d is 1, you have from your previous result that a - b = (n/d)r. That is, a = b mod n/d. But, you also have a = b mod n under the assumption f is 1-1. This forces d to be 1 using some kind of proof by contradiction. Or, a = b mod n/d suggests taking a to be 0 and b to be n/d originally, which also would lead to some kind of proof by contradiction, since you still have to rule out the case d = 1. In summary, your (backward) method of proof can lead you to a final proof, but will require some reworking to put it into a standard proof format. However, if you try the contrapositive, you get a better working hypothesis than you get with the direct method of proof. That is, with the contrapositive, you have the hypothesis that gcd(c,n) = d > 1. Your aim is to show that f is not 1-1. That is, you must find [a] and [b] such that [a] is not equal to [b] but [ca] is equal to [cb]. The analysis I gave above leads to choosing a = 0 and b = n/d. You just need to show a is not equal to b mod n. -- Bill Hale === Subject: sequence of continuous functions {f_n(x)} is a sequence of continuous functions on R^1 and it converges pointwise everywhere to f(x). Prove that for any G in R^1, f^-1(x) is the union of a countable collection of closed sets. Furthermore, let C_f be the set of continous points for f(x), show that C_f is dense in R^1. === Subject: Re: sequence of continuous functions > {f_n(x)} is a sequence of continuous functions on R^1 and it converges > pointwise everywhere to f(x). > Prove that for any G in R^1, f^-1(x) is the union of a countable ... for any x in R^1, f^-1(x) ... ??? > collection of closed sets. Furthermore, let C_f be the set of continous > points for f(x), show that C_f is dense in R^1. === Subject: Re: sequence of continuous functions Excuse me I meant for any G in R^1, f^-1(G) is the union of a countable collection of closed sets. === Subject: Re: Approximation by linear combinations of natural powers of $x$ > How does one prove that > $$ > min_{{a_k}} | 1 - sum_{k=1}^N a_k x^k |_2 > D e^{-cN}, > $$ > where $a_k$'s are any real numbers, $|cdot|_2$ is $L_2$ norm on > $[a,b]$, $00$, and $D>0$? Or, for those who don't take to TeX, How does one prove that there are positive constants D and c such that the L_2 norm on the interval [a, b] of a polynomial of degree N with real coefficients and constant term 1 is bounded below by D exp(-cN)? -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Proof against a corrupted math society <32tp61p6b1qfmiuqpmab2b0n16ilje62rq@4ax.com Let pq = 1313. > Decompose pq into it's prime factors p and q, *step-by-step* using your > polynomial-time aglorithm which you claim can achieve such. I'll > provide you a compensatory reward of $100 000 if you do it (seriously). I can see that you don't understand what I want. I don't care about money or legitimacy in the public's eyes. I want what any real mathematician would want: freedom to research without unnecessary distractions. That's what I want, and I'd prefer not to have publicity. It seems to me that if cooler heads prevail, this entire situation could be cleaned up with hardly anyone the wiser. Any knuckleheads who tried to break the status quo could simply be convinced that things are nicer the way they are. At this point in time, the situation is still contained, but that containment will not last long. James Harris === Subject: Re: Proof against a corrupted math society Mail-To-News-Contact: abuse@dizum.com >> polynomial-time aglorithm which you claim can achieve such. I'll >> provide you a compensatory reward of $100 000 if you do it >(seriously). >I can see that you don't understand what I want. >I don't care about money or legitimacy in the public's eyes. People posting statements that would decrease your legitimacy don't bother you, then. >I want what any real mathematician would want: freedom to research >without unnecessary distractions. I've found that Internet access in general, and Usenet in particular, are very distracting. You might want to sacrifice them for your work. >That's what I want, and I'd prefer not to have publicity. You probably ought to discontinue posting your results on Usenet in that case. -- Michael F. Stemper #include Visualize whirled peas! === Subject: Re: Proof against a corrupted math society >> Let pq = 1313. >> Decompose pq into it's prime factors p and q, *step-by-step* using >your >> polynomial-time aglorithm which you claim can achieve such. I'll >> provide you a compensatory reward of $100 000 if you do it >(seriously). >I can see that you don't understand what I want. >I don't care about money or legitimacy in the public's eyes. Of course you realize that this directly contradicts things you've said hundreds of times. Hint: You're the only mathematician I've ever seen with a web site titled mathforprofit, including a link where people can make donations by PayPal. >I want what any real mathematician would want: freedom to research >without unnecessary distractions. Huh? So stop posting on usenet and go ahead and do your research. I mean really, how has anyone been interfering with your freedom to do that? I can't come even close to figuring this one out. >That's what I want, and I'd prefer not to have publicity. What are people to make of this sort of statement, given the many times you've explained that we should all be fired because you're not famous? >It seems to me that if cooler heads prevail, this entire situation >could be cleaned up with hardly anyone the wiser. >Any knuckleheads who tried to break the status quo could simply be >convinced that things are nicer the way they are. >At this point in time, the situation is still contained, but that >containment will not last long. >James Harris ************************ David C. Ullrich === Subject: Re: Proof against a corrupted math society !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw [...] > Hint: You're the only mathematician I've ever seen with > a web site titled mathforprofit, including a link where > people can make donations by PayPal. Hey, my preview-latex project is highly interesting for mathematicians, and the old project page at SourceForge also included a PayPal donation link, and somebody even once gave US$25 through that link. If I had a cent for every download of that software, I'd have earned ten times that easily in the last year alone. I think that expecting donations from mathematicians is a far stretch even if you have something tangible to offer. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Proof against a corrupted math society >[...] >> Hint: You're the only mathematician I've ever seen with >> a web site titled mathforprofit, including a link where >> people can make donations by PayPal. >Hey, my preview-latex > project is >highly interesting for mathematicians, and the old project page at >SourceForge also included a PayPal donation link, Well sure, but the site wasn't named mathforprofit, was it? >and somebody even >once gave US$25 through that link. If I had a cent for every download >of that software, I'd have earned ten times that easily in the last >year alone. >I think that expecting donations from mathematicians is a far stretch >even if you have something tangible to offer. ************************ David C. Ullrich === Subject: Re: Proof against a corrupted math society Discussion, linux) > I can see that you don't understand what I want. > I don't care about money or legitimacy in the public's eyes. > I want what any real mathematician would want: freedom to research > without unnecessary distractions. > That's what I want, and I'd prefer not to have publicity. What happened to dreams of Oprah? Partying with heads of state? The commercial endorsements? You'd still take the chicks, right? > At this point in time, the situation is still contained, but that > containment will not last long. I wish I had a nickel for every time you said that. -- Jesse F. Hughes I am the barbarian at the gates. I am a revolutionary, a discoverer, a guy who didn't just try, but did, who didn't just wonder, but accomplished. -- James S. Harris gives Hollywood its tagline === Subject: Re: Proof against a corrupted math society <32tp61p6b1qfmiuqpmab2b0n16ilje62rq@4ax.com Let pq = 1313. > Decompose pq into it's prime factors p and q, > *step-by-step* using your polynomial-time aglorithm > which you claim can achieve such. I'll provide you > a compensatory reward of $100 000 if you do it > (seriously). > I can see that you don't understand what I want. > I don't care about money or legitimacy in the public's > eyes. > I want what any real mathematician would want: freedom > to research without unnecessary distractions. Like the need to solve practical problems like writing 1313 as 13 * 101. > That's what I want, and I'd prefer not to have > publicity. Then you really ARE a crank. If you don't want publicity about SF, then why did you ever CONSIDER posting on Usenet in the first place? Posting at Usenet lasts forever, and anyone with an ISP can access it. The very fact that you're posting means that you DO want publicity; this follows logically from the previous statement. > It seems to me that if cooler heads prevail, this > entire situation could be cleaned up with hardly > anyone the wiser. Except for the people who have ISP's and point their browsers at Google Groups Archive. > Any knuckleheads who tried to break the status quo > could simply be convinced that things are nicer > the way they are. Or, IF (and I'm using IF here) your results are true, any readers of your posts will be convinced that breaking the encryption of accounts is possible. IF they use your method, then that will make YOU an accessory to their crimes! Aiding and abetting a felony! YOU will wind up in jail. > At this point in time, the situation is still > contained, Nope. There's been enough time for your last post to circle the globe. Contained? Bullmuffins. > but that containment will not last long. Which means you'd better get ready for your jail time. --- Christopher Heckman === Subject: Re: Proof against a corrupted math society <32tp61p6b1qfmiuqpmab2b0n16ilje62rq@4ax.com Then you really ARE a crank. If you don't want publicity about SF, then > why did you ever CONSIDER posting on Usenet in the first place? Posting > at Usenet lasts forever, and anyone with an ISP can access it. Not on Google. Periodically the Drunken Raving phase is followed by the Remorseful Hungover phase, during which James systematically deletes all his posts from the Google archives. There are other archives of course, and also many of James' bon mots survive in the form of replies from other users. - Randy === Subject: Re: Proof against a corrupted math society <32tp61p6b1qfmiuqpmab2b0n16ilje62rq@4ax.com Let pq = 1313. Decompose pq into it's prime factors p and q, > *step-by-step* using your polynomial-time aglorithm > which you claim can achieve such. I'll provide you > a compensatory reward of $100 000 if you do it > (seriously). > I can see that you don't understand what I want. > I don't care about money or legitimacy in the public's > eyes. > I want what any real mathematician would want: freedom > to research without unnecessary distractions. > Like the need to solve practical problems like writing 1313 as 13 * > 101. > That's what I want, and I'd prefer not to have > publicity. > Then you really ARE a crank. If you don't want publicity about SF, then > why did you ever CONSIDER posting on Usenet in the first place? Posting > at Usenet lasts forever, and anyone with an ISP can access it. The very > fact that you're posting means that you DO want publicity; this follows > logically from the previous statement. You don't understand the real world. Most of what people know is what they're told to know. At this point this situation is easily handled, and those of you who might know more worldwide are easily handled, as what can you do? If you know enough to exploit the theorem, then eventually you'll make a move using it, and then you can simply be convinced that it's not a good idea to use it. In the real world, governments have power that few people care to think about, and it's not at all impossible for this situation to still be handled. And it's not difficult for any of you at any time to be convinced of the right course of action. James Harris === Subject: Re: Proof against a corrupted math society <32tp61p6b1qfmiuqpmab2b0n16ilje62rq@4ax.com> o, i know what you are doing crackpot. many people here have already said that your SFT can't lead to any useful factoring algorithm. you know this, but refuse to believe the truth. alas, you can't come up with an algorithm yourself, so you are hoping that someone might be stupid enough to try to come up with an algorithm which justifies the (non) importance of your SFT. fact is, nobody will be stupid enough, except you, to work on it because they know that SFT does not lead to a practically useful algorithm for factorization. === Subject: Re: Proof against a corrupted math society <32tp61p6b1qfmiuqpmab2b0n16ilje62rq@4ax.com Let pq = 1313. > Decompose pq into it's prime factors p and q, *step-by-step* > using your polynomial-time aglorithm which you claim can > achieve such. I'll provide you a compensatory reward of > $100 000 if you do it (seriously). What's the joke here? That $100 000 = $100 * 000 = $0? --- Christopher Heckman === Subject: Re: dumb stupid question (sorry) it here instead of sci.physics, also sorry that I used such a vague subject line -- heh, although, if I knew what to put in the subject === Subject: Re: Numerology is science, Cardinality is a myth > Surely with the will of 20 men you can take a stab at this? Guess not, since my will is not that great. === > Subject: Re: anyone want to see my numerology proof? > how can I put this, I'm in bare feet.... > A street (usenet) covered in broken glass has numerous houses > (newsgroups) and I run up to a house and start asking for help, > numerous expletives and I try the next house, numerous excuses and I > try the next, so I try the next, a big party full of people, and I'm > standing at the door shuffling around on broken glass, waving, waving, > day after day waving. > Who replied to me? > Ghost in the machine > William Elliot > Barb Knox > George Green > Gib Bogle I'll say Ghost, just because of the reference to glass. You seem to thrive on obscure references. Then again, when you go for the obvious (above) you miss. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Numerology is science, Cardinality is a myth <42700033$1_5@newsfeed.slurp.net> <42703703$1_2@newsfeed.slurp.net> what is the correct answer to my post, .....help I'm shufflling around outside a door standing on broken glass. WHAT SHOULD I DO? Herc === Subject: Re: Numerology is science, Cardinality is a myth > and The Truman Show is True. > and Bill Gates is the worlds most recognised BILLionaire, > Lady Di died, > HawKING is the smartest, > Tiger WOODs the best golfer, > Ronald Raegun initiated space ray guns, > Michelangelo painted angels, > TutanKham had the best tomb, > Nic Cage steals cars and goes to prison, > Peano formalised the unit number (phallus) > GODel formalised the unprovable, > Cantor can't order, > Britney SPEARS is cupid, > ... And I'm named after the patron saint of travel (St. Christopher, who was decanonized by the Catholic Church because there's no proof he existed), but I don't like any form of travelling. --- Christopher Heckman === Subject: Re: Numerology is science, Cardinality is a myth look you simpletons.. its not EVERYONES name. its most everyone who comes in contact with God incarnate (ME). if you've never talked to me in my life then I've never noticed you, so what is the point of using my macro quantum powers to coordinate things I will never see. or YOU HAVE TO MEET ME (for me to see you). What was your 1st reply to me about Chris? (actually bother if you can, its really pathetic you can't see 1 + 1 = 2 on the board in front of you} what was your 1st reply to me about Mr HECKMAN????? ITs *****MY***** power not yours. You're just the puippets at the end of my strings. the post PLEASE MR HECKMAN Herc === Subject: Re: Numerology is science, Cardinality is a myth >look you simpletons.. >ITs [sic] >*****MY***** power not yours. You're just the puippets [sic] >at the end of my strings. You wish. If we're all your puppets then how come we all think you're a LOON? Maybe you WANT to be thought of as a loon as part of your grand plan? If not, then it sure looks like the puppet master's powers are just not up to the job. Impotent twerp. -- --------------------------- | BBB b Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | ----------------------------- === Subject: Re: Numerology is science, Cardinality is a myth times are changing... it was written in your name all your life, that your peculiar desitny was only to answer this very post... how can I put this, I'm in bare feet.... A street (usenet) covered in broken glass has numerous houses (newsgroups) and I run up to a house and start asking for help, numerous expletives and I try the next house, numerous excuses and I try the next, so I try the next, a big party full of people, and I'm standing at the door shuffling around on broken glass, waving, waving, day after day waving. Who replied to me? Ghost in the machine William Elliot Barb Knox George Green Gib Bogle THE ANSWER IS KNOCKS. to evade BARBS. Game over Barb, your immortality vanished the nanosecond you hit that 1st Reply button to me. Your various charades of ignorance are coming to an end. People in aus.tv have HAIL HERC THE LIVING LEGEND in their sigs, and I'm flying up more to witness how the US govt. makes a show of my life. Herc === Subject: Re: Numerology is science, Cardinality is a myth >times are changing... it was written in your name all your life, that >your peculiar desitny was only to answer this very post... >how can I put this, I'm in bare feet.... >A street (usenet) covered in broken glass has numerous houses >(newsgroups) and I run up to a house and start asking for help, >numerous expletives and I try the next house, numerous excuses and I >try the next, so I try the next, a big party full of people, and I'm >standing at the door shuffling around on broken glass, waving, waving, >day after day waving. >Who replied to me? >Ghost in the machine >William Elliot >Barb Knox >George Green >Gib Bogle >THE ANSWER IS KNOCKS. to evade BARBS. Game over Barb, your >immortality vanished the nanosecond you hit that 1st Reply button to >me. Your various charades of ignorance are coming to an end. >People in aus.tv have HAIL HERC THE LIVING LEGEND in their sigs, and >I'm flying up more to witness how the US govt. makes a show of my life. In case anyone cares, here's the actual initial exchange: >> how can I put this, I'm in bare feet.... >> A street (usenet) covered in broken glass has numerous houses (newsgroups) >> and I run up to a house and start asking for help, >Specifically what sort of help are you seeking in sci.logic? If you have >some (allegedly) logical argument you want analysed then just post it and >ask for analysis. >> numerous expletives and >> I try the next house, numerous excuses and I try the next, 'knocking is rude' >> so I try the next, a big party full of people, and I'm standing at the door >> shuffling around on broken glass, waving, waving, day after day waving. >Maybe you should try alt.obscure.analogies... And are you going to deign to reply to the my most recent post? -- >>look you simpletons.. >>ITs >[sic] >>*****MY***** power not yours. You're just the puippets >[sic] >>at the end of my strings. >You wish. If we're all your puppets then how come we all think you're a >LOON? Maybe you WANT to be thought of as a loon as part of your grand plan? >If not, then it sure looks like the puppet master's powers are just not up >to the job. Impotent twerp. -- --------------------------- | BBB b Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | ----------------------------- === Subject: Re: Numerology is science, Cardinality is a myth so? your name is very obviously linked to your FIRST reply to me. why don't you admit it you simpleton?. you point out all the obscure matches and make jokes then you ignore the dozens of extemely highly correlated name by nature matches. why don't you let Mr HECKMAN post up what his 1st reply to me was? ON THE BOARD. OBVIOUS. as for your other garbage, you think I'm a loon because I make grand claims, my grand claims can't be true because you think I'm a loon. catch 22. Truth is you're a 60 year old spinster lecturer with deluded claims to purile archaic maths. You all swear black and blue the Halt hypothesis is irrefutable, then I prove that you can program a halt function that outputs for any program a, Program (a) halts with probability 99.999999999%. For as many 9s as you wish to run the program for. Then you ignore the fact oo people could come up with oo sequences and every combination is covered to_oo_length, and you STILL act benign when I point of Russels Set in your hailed powerset proof. This is the stuff of year 10 applied maths and you STILL don't get i! And you STILL claim YOU CANT PROVE ME is some magic formula that refutes the very idea of completeness. 19th and 20th century maths was all BOGUS! They didn't have computers, they didn't have infinte stream handling technology, they didn't have AI and the enumerated computer funcitons they did dabble with was entirely naive. You're old Barb, there's nothing God can write to you. Empty your mind, imagine you're in a new universe, take no laws for granted and reread the thread. I'm making rhymes of every famous name in history and every name here dictates what you think. This is your powerset proof in English. For any given set, the Powerset is the set of all subsets. One such subset is the set of all elements that don't contain themselves in the mapping from the set to the powerset. Therefore no mapping exists. Russels Set? Herc === Subject: Re: Numerology is science, Cardinality is a myth You're obviously out of your depth here. Herc has strong and valid >opinions about logical and philosophical issues you probably aren't >even aware exist. He is also mentally ill. The two have none to do >with each other, and he still deserves respect. >'cid ... What was I thinking!? 'cid 'ooh === Subject: Re: Numerology is science, Cardinality is a myth sci.logic ~ 3 George's and the fiery Dragon but did my sword strike hard enough? Herc === Subject: Re: Numerology is science, Cardinality is a myth <42700033$1_5@newsfeed.slurp.net And what, pray tell, do you make of my name? > Will Twentyman > email: wtwentyman at copper dot net Need full name and birthdate for that. -Stacey internetstuff47129@yahoo.com === Subject: Re: Numerology is science, Cardinality is a myth >>And what, pray tell, do you make of my name? >>Will Twentyman >>email: wtwentyman at copper dot net > Need full name and birthdate for that. > -Stacey > internetstuff47129@yahoo.com I've never seen evidence that Herc feels a need for either. -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Numerology is science, Cardinality is a myth <42700033$1_5@newsfeed.slurp.net And what, pray tell, do you make of my name? > Will Twentyman > email: wtwentyman at copper dot net Need full name and birthdate for that. -Stacey internetstuff47129@yahoo.com === Subject: Types of functions and relations I seek to understand types of functions and the relation properties that they obey, if there is any at all. Please add to the list below other types of functions. It seems that designating a type of function means that such functions have interesting properties, such as integrals over even and odd functions. Note, I do not include specific groups of functions such as trigometric, exponential, etc, because I seek properties that any function might have. Composite. Even and Odd. Inverse. Degrees: linear, quadratic, cubic,etc. Monotonic. Periodic. Constant. Injective. Surjective. Bijective. What is the name of the set whose elements constitute the pairs that the function maps to. for instance, let f: A -> B be a function with elements of the set F <= AxB. What is the formal name of F? === Subject: Re: Types of functions and relations [...snip...] > What is the name of the set whose elements constitute the pairs that the > function maps to. for instance, let f: A -> B be a function with > elements of the set F <= AxB. What is the formal name of F? You may possibly have a misunderstanding here. A common terminology is: A is the 'domain' of f B is the 'codomain' of f (some say 'target' instead of 'codomain') The function f maps elements of A to elements of B (not to pairs in AxB as you state). If 'a' is in A and f(a)=b, then 'b' is called the image of 'a' under f, and 'a' is a pre-image of b. The set of all images of elements of A under f is a subset of B (possibly all of B) called the 'range of f'. The subset F you describe above of all pairs (a,b) such that 'a' is in A, 'b' is in B and f(a)=b is commonly called the 'graph' of f. Not all books define all these terms in the same way, and this causes some confusion and misunderstanding. -- Jim Buddenhagen === Subject: Re: Types of functions and relations > You may possibly have a misunderstanding here. A common terminology is: > A is the 'domain' of f > B is the 'codomain' of f (some say 'target' instead of 'codomain') That is my understanding too. > The function f maps elements of A to elements of B (not to pairs in AxB > as you state). If 'a' is in A and f(a)=b, then 'b' is called the image > of 'a' under f, and 'a' is a pre-image of b. The set of all images of > elements of A under f is a subset of B (possibly all of B) called the > 'range of f'. Understood. What I meant was that a function can be seen as a set whose elements are certain pairs from AxB. Not all of them, just some. When we say that f(a) = b, it means that the pair (a,b) is an element of F<=AxB. I did not know what to call such a set. > The subset F you describe above of all pairs (a,b) such that 'a' is > in A, 'b' is in B and f(a)=b is commonly called the 'graph' of f. Okay. I will call it the graph until further notice. === Subject: Re: Types of functions and relations > I seek to understand types of functions and the relation properties that > they obey, if there is any at all. > Please add to the list below other types of functions. It seems that > designating a type of function means that such functions have > interesting properties, such as integrals over even and odd functions. > Note, I do not include specific groups of functions such as trigometric, > exponential, etc, because I seek properties that any function might have. > Composite. > Even and Odd. > Inverse. > Degrees: linear, quadratic, cubic,etc. > Monotonic. > Periodic. > Constant. > Injective. > Surjective. > Bijective. Continuous. Differentiable. Differentiable with continuous derivative. Twice Differentaible. Twice differentiable with continuous second derivative. etc. Uniformly continuous. Integrable. Square integrable. p-th power integrable. Of bounded variation. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Types of functions and relations > Continuous. Differentiable. Differentiable with continuous derivative. > Twice Differentaible. Twice differentiable with continuous second > derivative. etc. Differentiable with continuous derivative seems to be two types of functions applied to two different functions; the differentiable one, and the result of the derivative. True? > Uniformly continuous. So a function can be continuous in a region, or uniformly continuous? > Integrable. Square integrable. p-th power integrable. > Of bounded variation. What is of bounded variation? === Subject: Re: Types of functions and relations > Continuous. Differentiable. Differentiable with continuous derivative. > Twice Differentaible. Twice differentiable with continuous second > derivative. etc. > > Differentiable with continuous derivative seems to be two types of > functions applied to two different functions; the differentiable one, > and the result of the derivative. True? It refers to a single function f which has a continuous derivative (as opposed to a function g which has a discontinuous derivative). > Uniformly continuous. > > So a function can be continuous in a region, or uniformly continuous? No, that's not what uniformaly continuous means. It's a concept you most likely won't see in an intro calculus course, but in a higher level undergrad analysis course, or in graduate-level real analysis. > Of bounded variation. > What is of bounded variation? Another concept from analysis-beyond-calculus. There are plenty of analysis textbooks that will supply you with definitions, or you can just put your faith in Google. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: SF: Generalized SFT's [Proginoskes] [...] >> from a post made in 1994, not by James Harris, but by >> Alexander Abian, a certified crank. Hey, Abian is closer to my generation: don't even think about putting them in the same sentence. Professor Abian was unfailingly civil, and while he was indeed a world-class crank when it came to physics, he was also a respected mainstream mathematician. Here's a list of 250+ of his published math papers: http://www.math.ucdavis.edu/~suh/abian/abian-list.html God only knows why he went on (& on, & on) about the necessity for blowing up the evil moon (one of his more, umm, colorful physics crusades), but some of his pure math posts on Usenet were models of patience, precision and clarity. He was an extraordinary character, and I miss him. RADICAL CHANGE IS ABSOLUTELY NECESSARY. THE MOST PRACTICAL THE MOST EFFECTIVE THE MOST URGENT RADICAL CHANGE IS TO BLOW UP THE MOON TO SMITHEREENS AND GET RID OF ITS EVIL PRESENCE ONCE AND FOR ALL! Yup, I even miss that . === Subject: Re: SF: Generalized SFT's [...] >> from a post made in 1994, not by James Harris, but by >> Alexander Abian, a certified crank. > Hey, Abian is closer to my generation: don't even think about > putting them in the same sentence. Actually, I only mention Abian because James Harris's posts reminded me of something he once said, which I ran across due to a pinball project blowing up the moon.) There's a program called Visual Pinball (available for free at http://www.randydavis.com/vp/ ), which uses Visual Basic as a scripting language, and it allows you to build a pinball table, with kickers, plungers, bumpers, ramps, etc. You can also import graphics and sounds, and some people have made some really nice tables; others have taken existing tables and simulated them. Someone posted a thread at Visual Pinball Forums called What would the Internet look like, if it was a pinball machine? Someone had described in detail a fictitious pinball table about ten years ago, and mentioned a lot of cranks at the time, and the whole table was called Net.Nuisances.Pinball. Well, working on the description, I brought it into actuality. It's gone through several versions, currently at version 0.4. If you want to take a look at it, you can go to my pinball page, at http://www.public.asu.edu/~checkma/pinball/ . --- Christopher Heckman === Subject: Re: Numerical evaluation of a contour integral <24757400.1114604141267.JavaMail.jakarta@nitrogen.mathforum.org> yeah, the 'quadl' function in Matlab is based on Lobatto's adaptive method, and it doesn't seem to work so well for the functions I am trying to integrate. === Subject: Re: Numerical evaluation of a contour integral >yeah, the 'quadl' function in Matlab is based on Lobatto's adaptive >method, and it doesn't seem to work so well for the functions I am >trying to integrate. Can you give us an example of the sort of function you have in mind? Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Numerical evaluation of a contour integral <24757400.1114604141267.JavaMail.jakarta@nitrogen.mathforum.org> method, and it doesn't seem to work so well for the functions I am >trying to integrate. > Can you give us an example of the sort of function you have in mind? the integration can be done by residues, but I have to the integrals of many functions (in a recursive algorithm), so i would like to have it computed numerically. not all the functions are difficult to integrate but at a particular step there can appear functions such as: a(z)/b(z)=(1+2z)(1+0.5z)/((z-0.99)(z-1/0.99)(z-1.09 e^{i3})(z-1/1.09 e^{-i3})(z-1/1.05 e^{-i2})(z-1.05e^{i2}) where $b(z)$ can have roots near the unit circle (simultaneously outside and inside the unit circle such as in the example above). === Subject: Re: Numerical evaluation of a contour integral Hi Jay, by using the method outlined previously (solving the ordinary differential equation dI/dt = a(e^(it))/b(e^(it))*i*e^(it), I(0)=0 in the range [0,2*pi]), I get I(2*pi) = 5.7076144875 - 125.39799267 i as the value for the contour integral. If the Lobatto quadrature doesn't work well, maybe you should try a solver for ordinary differential equations in MATLAB. Best wishes Torsten. === Subject: Re: Numerical evaluation of a contour integral > i wish to get a numeric value for the integral of a rational complex >function over the unit circle C= {z in R : |z|=1}. so the function >is f=a/b where a and b are polynomials, and b has no roots on C. >However, b can have roots close to C. are there good numerical >algorithms for computing this integral? If b has roots near C, I would suggest, if possible, moving the contour away from those roots (either not passing over any others, or taking into account the residues of those you do pass). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Differentiability at a point: understanding starting from set theory; request for help. > Instead of defining some sequence getting closer and closer to some > value in terms of a metric, we define closer and closer as the sequence > being contained in some sequence of subsets (o_1, o_2, o_3, ...) where > each subset is a subset of the previous one. Then wouldn't each sequence be generated for each limit? > The metric is a special case of this type of behaviour. Consider a > subset of R which consists of all points that are closer than some > distance d to some point L. Then a sequence of points (p_1, p_2, p_3, > ...) that are succesively closer to the point L, can be thought of as a > sequence of subsets of the points that are as close to L as p_1, p_2, > etc. > When you don't have a metric, you could still have the similar > situation of subsets-nested-in-subsets, and the limit will be the > intersection of all those subsets (assuming it exists). I'll have to read more about this, as I don't fully understand it. > Derivatives don't always exist, even on regular functions. Think of > the function f, were f(0) = 1, and f(x) = 0 for all x <> 0. This has no > derivative at 0. There are other functions (non-continuous functions) > that have no derivative at an infinite number of points. Understood; derivatives don't always exist. > Usually, a derivative (or differential) function will be defined on > some domain of functions which are continuous, or smooth in some > sense. Okay. > Once you start down the rabbit hole, there's no end :) I'll paraphrase that for a signature, lol. Quite true, quite true! > If you're mostly interested in seeing how derivatives on the reals > really work, you might pick up a book on Analysis. People here seem > to recommend Rudin's Principles of Mathematical Analysis, which is > most likely at your local library. The texts that I have on calculus are: Calculus of Several Variables, Adams. Calculus: Single variable, early transcendentals, Stewart. Calculus: A first course, Stewart. Variational methods in optimization, Smith. Schaum's Differential Equations. Schaum's Advanced Calculus. Fundamentals of Complex Analysis for Mathematics, Science, and Engineering, Saff and Snider. As well as texts on solving differential equations. The reason why I'm reviewing calculus and seeking a deeper understanding of it is because I attempted to learn the calculus of variations from the text listed above and found it difficult. The text bases everything on vector spaces and limits and I discovered that I didn't know those topics deeply enough. === Subject: Re: Differentiability at a point: understanding starting from set theory; request for help. > Yes. But that kind of metric is derived from something called a norm, > and that's what you really need to understand. For the rational > numbers, the only norms that have nice properties are the real norm > (the absolute value you see above), and the various p-adic norms, a > different one for each prime integer p. In general, if the norm is > called n, and the distance metric is called d, then d(x,y) is defined > as n(x-y). Given the rational (not real) numbers, you can pick any of > these various norms, and with your choice you can then define the > distance metric, and then you can take the set of equivalence classes > of Cauchy sequences and voila you have the appropriate completion of > the rationals, namely the reals or one of the p-adic fields. I've learned a little about abstract norms in my study of linear algebra, specifically, vector spaces. There seem to be parallels here as well, although I've not learned yet what p-adic functions are. > Now in elementary calculus, which deals with the reals, you've already > completed the rationals using the real metric derivived from the real > norm, so you're stuck taking limits and derivatives using that > metric/norm instead of one of the p-adic metrics/norms. Ah! That is some understanding that I was after! As I understand it, I would need to go further back in the definitions outside of calculus, to the very concept of real numbers, in order to apply a meaningful change in the metric. > Another example: If the value of the norm is always an integer, and > another property is satisfied (can you guess what it is?), then the > topology is exactly the integers under the real norm. That the value of the norms is ordered? > Actually you can generalize that: If the distance metric satisfies the > flat-triangle condition (I made up that term, guess the obvious > meaning), then the space is totally ordered (in either of two ways like > before). Well, is the flat-triangle condition that the abstract space has a constant metric at all points and that it is nothing but the normal Euclidean one? where the distance equals sqrt((x1-x2)^2 + (y1-y2)^2 + (z2-z1)^2) implies sum of angles in triangle is 180 degrees? I'm new to pure math and definitions, etc. > Define B between A and C if d(A,B) + d(B,C) = d(A,C). > If for all points A,B,C, at least one of the points is between the > other two, then this betweenness property can directly generate a total > ordering, as soon as you've made one arbitrary binary decision. > (If you haven't guessed, betweenness and flat-triangle are same thing.) > Can you see the trivial definition of the total ordering? Yes, I see how that definition if modified to an iff would work. It seems to the normal intuitive understanding of what it means for something to be between two others. All elements could be put into a list using the betweenness property, however, in order for it to be an ordering the first (or last) element would need to be specified. That would, in my mind, rotate the list around to whichever way you wanted. I think visually, so that is how I see it in my mind. Is that what you meant? === Subject: Every square integrable function has integral representation? My space is L2 on compact interval [a,b]. Is it possible to write down any function in L2, say g(x),with intergral form? Like g(x) = integral a to x f dm. === Subject: Re: Every square integrable function has integral representation? > My space is L2 on compact interval [a,b]. > Is it possible to write down any function in L2, say g(x),with > intergral form? Like g(x) = integral a to x f dm. g(x) = 0 if x irrational = 1 if x rational g(x) = intergral(a,x) ???? dm === Subject: Re: Every square integrable function has integral representation? >> My space is L2 on compact interval [a,b]. >> Is it possible to write down any function in L2, say g(x),with >> intergral form? Like g(x) = integral a to x f dm. >g(x) = 0 if x irrational > = 1 if x rational >g(x) = intergral(a,x) ???? dm Bad example: this is 0 almost everywhere (and a member of L^2 is really an equivalence class under almost everywhere equality). But try the indicator function of a fat Cantor set: a nowhere dense set of strictly positive measure. The functions that have (almost everywhere) an integral representation g(x) = g(a) + int_(a,x] dmu with mu a signed measure are (almost everywhere equal to) functions of bounded variation. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Every square integrable function has integral representation? > The functions that have (almost everywhere) an integral representation > g(x) = g(a) + int_(a,x] dmu with mu a signed measure are (almost everywhere > equal to) functions of bounded variation. and the functions that have (almost everywhere) an integral representation g(x) = g(a) + int_a^x f(x) dx are (almost everywhere equal to) absolutely continuous functions. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: difficult problems for us mathematicians. 1. Do you know of people who are designing a plane? Or who are designing a missile to go to Mars? Do you know they have to think of sudden breakthroughs and what can happen then. 2. Do you know people are dealing with sand supplementary and that moves with the water but still has to function. Do you know that that is difficult. Did you ever see a pan with boiling water and in it an rough variable pattern on the surface and in the water. Are you ready to make a program, so a software simulation. Did you ever stand at a beach see that water and making an exact computer program from that water with small and later when the wind grows with huge waves. Sometimes with white heads. And then the breakers. Breakers that are shuddering in a white mist? So nice to make exact? Do you know that stars and the moon can reflect in those moving waves. Can you make that exact. And the planets and stars move along the sky and also that is interesting. The moon has a structure that reflects in the sea. Oh boy what can we do now as mathematician. And then we are there, what a great theater. I feel romance. Could you make romance exact? ------------------------------- The railway station a. High you are standing and you are looking down on the station hall and you see dots. many dots for this is a busy time. with a lot of people down there. And the move like small self-decoders. Each with an own will. And with hardly any coherence. Disorder. Physical chaos. b. Wow! You see a circle arising from that disorder. Circles we can recognize, for sure. But a moment later the circles have disappeared and the disorder is back. c. All of a sudden you see lines through the disorder. Clearly lines. Also to recognize. d. A bell is sounding. It becomes later but your eyes work well, and you look again. And now? Everything has gone. All dots are gone. This station is now empty of people. e. Nothing else to see here. And therefore you go home. It's dark outside. Is statistics possible? What do you think? What can we do with that disorder with still some structure for the density is also a disorder. The density changes in time and place and is also in disorder. And this is reality. At some places people are standing close to each other. Some are in love. Some groups are available. And we can't foretell what will happen. All happens so suddenly. People came from the outside. Like a thief in the night is coming. Unexpected. And what do we here with common mathematics? Please play with mathematics. And still eveywhere are mathematicians needed. ed === Subject: Re: difficult problems for us mathematicians. > Did you ever see a pan with boiling water and in it an rough variable pattern on the surface and in the water. Are you ready to make a program, so a software simulation. In a sense! When cooking perogees (potatoes wrapped in pasta/pastry) I stirred the water in a clockwise direction. Upon ceasing to stir, the perogees would all rotate a the same rate and maintain the same relative orientations to each other. However, after a given time the water would slow down to a point where the perogees would instantly break the pattern and begin to move independently of each other. It seemed that the system went from predictable to chaotic when the rotational velocity got too low. It was very interesting to repeat the effect over and over. Almost made me wish the food took longer to cook. Adam. === Subject: Re: Noosphere Academy: An absolute NA's ignoramus judges about a top world's professional <6s87$21f83t8.47782a6$59n1@nah Why don't you guys switch to Russian and spare us all of > this kooky exchange? ... aren't they all the same person ? === Subject: Re: Ancient Greeks of Euclid never used reductio ad absurdum; Infinitude of Primes is a direct proof <426DF3A4.653CEC2C@iw.net> <426E8AC4.539C6298@iw.net> that is exactly what it says in the Clark U. version: two branches, G prime or not, one of which is absurd -- see following post. thus: anyway, the two parties virtually merged in Dec.99, with the Financial Services Modernization Act, thus begetting the phony 527 cmtes, run by the likes of Nixon hacks and Soros -- not the idiot ideologs of the DLC and their republican counterparts (as if they know the meaning of the word, republic !-) thus: b) the hare-brained analysis of WTC7 assumes that the substructure of the complex, which included a large subway station, was not affected by the two largest buildings, falling down into it. --Chair Man George NAND Strep Throat @ W'gate? http://tarpley.net/bush12.htm http://laroucehpub.com=20 http://members.tripod.com/~ame[CapitalEth]=ADrican almanac === Subject: JSH: The Quadratic that Ate Manhattan national security implications. It has no implications whatever. Forget that and solve one of the following little problems. It shouldn't take you more than a week, and each one is worth US$5,000,000 (in any desired currency). http://www.rewardsforjustice.net/english/wanted_captured/index.cfm?page=abde rraouf http://www.rewardsforjustice.net/english/wanted_captured/index.cfm?page=fake r http://www.rewardsforjustice.net/english/wanted_captured/index.cfm?page=ali_ sayyid_al-bakri http://www.rewardsforjustice.net/english/wanted_captured/index.cfm?page=Midh at_Mursi http://www.rewardsforjustice.net/english/wanted_captured/index.cfm?page=Nasa r === Subject: Re: SF: Symmetry, practical matters, social realities <9a2dneL0LKnp9_PfRVn-sg@comcast.com> wow; second roots aren't needed? thus: anyway, the two parties virtually merged in Dec.99, with the Financial Services Modernization Act, thus begetting the phony 527 cmtes, run by the likes of Nixon hacks and Soros -- not the idiot ideologs of the DLC and their republican counterparts (as if they know the meaning of the word, republic !-) thus: b) the hare-brained analysis of WTC7 assumes that the substructure of the complex, which included a large subway station, was not affected by the two largest buildings, falling down into it. --Chair Man George NAND Strep Throat @ W'gate? http://tarpley.net/bush12.htm http://laroucehpub.com http://members.tripod.com/~american_almanac === Subject: Re: SF: Symmetry, practical matters, social realities <9a2dneL0LKnp9_PfRVn-sg@comcast.com> so, why'd he use second roots, if they are completely unneccesary -- any use at all for them? >b) the hare-brained analysis of WTC7 assumes that the substructure of the complex, which included a large subway station, was not affected by the two largest buildings, falling down into it. --Chair Man George NAND Strep Throat @ W'gate? http://tarpley.net/bush12.htm http://laroucehpub.com http://members.tripod.com/~ame[CapitalEth]rican almanac === Subject: Writing calculus notes, order of limit definitions In order to better my understanding of calculus I am going to write notes on it in the Latex document format. The sequence of definitions should follow a logical order beginning with the definition of a limit and proceeding all the way to the fundamental theorem of calculus, where differential calculus and integral calculus are related. Since I will be using limits as the basis for the calculus, my first There is the definition of a full limit, and then a limit from the left and from the right. My text introduces the full version before that of the left and right, but the full version doesn't seem as basic as the left and right definitions since if a function has a limit from the left and from the right and they are equal, then the full limit exists. Should I define the left and right limits and then the full limit, or the full limit and then the former? The definitions are below. Definition. Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is L, and we write lim_{x -> a} f(x) = L if for every number epsilon > 0 there is a corresponding number delta > 0 such that |f(x) - L| < epsilon whenever 0 < |x - a| < delta. Definition of left-hand limit. lim_{x -> a from left} f(x) = L if for every number epsilon > 0 there is a corresponding number delta > 0 such that |f(x) - L| < epsilon whenever a - delta < x < a Definition of right-hand limit. lim_{x -> a from right} f(x) = L if for every number epsilon > 0 there is a corresponding number delta > 0 such that |f(x) - L| < epsilon whenever a < x < a + delta. Also, would anyone like to keep me on track with the calculus notes by reading them every now and then? Adam. === Subject: Re: Writing calculus notes, order of limit definitions : Since I will be using limits as the basis for the calculus, my first : There is the definition of a full limit, and then a limit from the : left and from the right. My text introduces the full version before : that of the left and right, but the full version doesn't seem as basic : as the left and right definitions since if a function has a limit from : the left and from the right and they are equal, then the full limit exists. : Should I define the left and right limits and then the full limit, or : the full limit and then the former? The definitions are below. One reason for giving the full limit separately (as opposed to in terms left and right limits) is that the definition carries over to other spaces (e.g., R^n, or suitably modified, to general metric spaces). The statement that the full limits iff the left and right limits exist is special to R and does not generalize to other spaces. : Also, would anyone like to keep me on track with the calculus notes by : reading them every now and then? Do you mean notes you post here, like in this post? Ted === Subject: Re: Writing calculus notes, order of limit definitions > One reason for giving the full limit separately (as opposed to > in terms left and right limits) is that the definition carries > over to other spaces (e.g., R^n, or suitably modified, to general > metric spaces). single real variable. I suppose I should use the more abstract definition of a limit that is valid for any R^n. > The statement that the full limits iff the left and right limits exist > is special to R and does not generalize to other spaces. I did not know that, but it seems to make sense since other spaces can be approached from many different directions. > : Also, would anyone like to keep me on track with the calculus notes by > : reading them every now and then? > Do you mean notes you post here, like in this post? Once I have the proper, logical order of the definitions and the theorems, I will type them in the Latex format which would be used to create a PDF file. I can post snippets here, in ASCII, if that is best for people to aid me in my project. Would be willing to help out? === Subject: Linear algebra question I have this question on a practice test: If eigenvalues of a matrix are positive and real, can the matrix be singular? Can anyone explain this to me please? === Subject: Re: Linear algebra question <6065430.1114662578316.JavaMail.jakarta@nitrogen.mathforum.org>, > I have this question on a practice test: > If eigenvalues of a matrix are positive and real, can the matrix be singular? > Can anyone explain this to me please? A square matrix, A, must have a zero eigenvalue to be singular. If Ax = 0 for any non-zero vector x, then 0 is the eigenvalue of that vector. If Ax is not a zero vector for every non-zero vector x (of appropriate dimension), A will be invertible. Can you show that if A maps a linearly independent set of vectors into a dependent set, then it maps a non-zero vector to the zero vector? === Subject: Re: Hilbert spaces I'll look for the book... the title seems quite promising. === Subject: Vector spaces and calculus To my understanding, the benefit of the concept of a vector space is that there are many mathematical ideas that share common properties, and by proving theorems involving vector spaces one also has proved the theorems for the many concepts that fit vector spaces. True? Is there any labor savings or benefit to using ideas of vector spaces in the study of calculus? I read online that the set of continuous functions is a vector space, and that the set of derivatives is also a vector space since the sum of two derivatives is also a derivative. True? Now, when studying calculus what parts can be generalized to vector spaces, or what theorems concerning vector spaces can I apply to derivatives and integrals? I seek a labor savings and a more abstract understanding of the calculus ideas, as it became quite apparent that I lacked a solid understanding after attempting to learn the calculus of variations. === Subject: Re: Vector spaces and calculus In linear algebra one studies among other things linear function f:V-->W, where V,W are vector spaces. Note that you can not even define the notion of a linear function without (at least implicitely) talking about vector spaces. In calculus one studies differentiable functions f: A --> B, where A and B are subsets of the reals, of n-dimensional space over the reals, or in the most general case of a normed vector space. Differentiability of f at a point x means: approximate the function f near x by a linear function. So linear algebra is a prerequisite of calculus. H === Subject: Re: Vector spaces and calculus > In linear algebra one studies among other things > linear function f:V-->W, where V,W are vector > spaces. Note that you can not even define the notion > of a linear function without (at least implicitely) > talking about vector spaces. This is the definition of a linear transformation that I learned. It must be similar to the one you use, yes? Let V, W be two vector spaces. A linear transformation (or linear map) from V to W is a transformation T: V --> W such that for all vectors u and v of V and any scalar c 1) T(u + v) = T(u) + T(v); 2) T(cu) = cT(u). > In calculus one studies differentiable functions > f: A --> B, where A and B are subsets of the reals, > of n-dimensional space over the reals, or in the > most general case of a normed vector space. The normed vector space is a vector space with the definition of an inner product? > Differentiability of f at a point x means: approximate > the function f near x by a linear function. > So linear algebra is a prerequisite of calculus. I've never really thought of it that way. In a preface of a text I have, the author mentions that calculus should be taught along with linear algebra because they are so related and one would have a better appreciation for both subjects if one did so. That's partly why I am seeking to understand the relationship between linear algebra/vector spaces and calculus. === Subject: Re: Vector spaces and calculus There's lots of places where linear algebra overlaps with calculus. The least-squares problem can be formulated as a linear system. In fact, it can be shown to be equivalent to the orthogonal projection onto a linear subspace of n-th degree polynomials. The Implicit Function theorem is essentially linear algebra; the solutions of the partial derivatives can be calculated using Cramer's rule. The Jacobian of the system of functions is the determinant of a matrix! Differentiation and integration are linear transformations. N-th degree polynomials can be seen as a vector space, and differentiation as a linear transformation to the vector space of (N-1)-th degree polynomials (try it!). Solutions to homogeneous differential equations can be seen as linear subspaces. In fact, a differential equation can be seen as a linear equation with a derivative operator. The Laplace transform is where linear algebra and calculus/analysis have common boundaries. The Fourier transform is a linear transformation on the space of, say, 1-d signals (ie., vectors!); the Discrete Fourier Transform is a matrix multiply! The list goes on... === Subject: Re: Vector spaces and calculus > There's lots of places where linear algebra overlaps with calculus. > The least-squares problem can be formulated as a > linear system. In fact, it can be shown to be equivalent to the > orthogonal projection onto a linear subspace of n-th degree > polynomials. > The Implicit Function theorem is essentially linear algebra; the > solutions of the partial derivatives can be calculated using > Cramer's rule. The Jacobian of the system of functions is > the determinant of a matrix! > Differentiation and integration are linear transformations. N-th > degree polynomials can be seen as a vector space, and differentiation > as a linear transformation to the vector space of (N-1)-th degree > polynomials (try it!). > Solutions to homogeneous differential equations can be seen as > linear subspaces. In fact, a differential equation can be seen > as a linear equation with a derivative operator. The Laplace transform > is where linear algebra and calculus/analysis have common > boundaries. > The Fourier transform is a linear transformation on the space > of, say, 1-d signals (ie., vectors!); the Discrete Fourier Transform is > a matrix multiply! > The list goes on... Wow, that is a very nice post. I had no idea of most of those relationships. The Jacobian formula seems to appear in many places. Poisson brackets comes to mind, but my prof never mentioned the Jacobian when deriving and defining them. Is there a benefit to seeing the differentiation and integration as linear transformations? Besides the implicit function theorem, what theorems from vector spaces are of great benefit in their application to the study of calculus? I'm interested in theorems that of a importance. === Subject: Re: Vector spaces and calculus > Is there a benefit to seeing the differentiation and integration as > linear transformations? Oh yes! The consequences of this and the stuff you can do with it runs very very deep into the realms of functional analysis, and a lot of other fields. Numerically solving difficult differential or integral equations (or perhaps a whole system of them) boils down, most of the time, to solving a linear system of equations. I can't even begin to describe all or even part of it, mostly because I don't yet understand a fraction of it myself. But look up Laplace transforms, Finite Element methods. The use of linear algebra breaks down a bit when we start talking about infinite dimensional vector spaces, and most of the time differential equations involve functions that are elements of such spaces. > Besides the implicit function theorem, what theorems from vector spaces > are of great benefit in their application to the study of calculus? I'm > interested in theorems that of a importance. I bet that you can randomly choose any fundamental theorem of linear algebra and the chances are big that you can use them somehow in solving calculus problems. Most of the time, the fundamental theorems are used to prove more advanced theorems which are then subsequently used in other fields (such as calculus). The Rank-Nullity theorem is such a fundamental theorem that is used in proving many other results. Willem === Subject: Re: Vector spaces and calculus > Oh yes! The consequences of this and the stuff you can do with > it runs very very deep into the realms of functional analysis, and > a lot of other fields. Numerically solving difficult differential or > integral > equations (or perhaps a whole system of them) boils down, most > of the time, to solving a linear system of equations. I can't even > begin to describe all or even part of it, mostly because I don't yet > understand a fraction of it myself. But look up Laplace transforms, > Finite Element methods. I've studied basic finite-difference methods when learning numerical methods for computer modeling. In it, I used matrix algebra, and thus linear algebra, to solve systems of equations, as you stated. However, when I think of calculus, I do not think of the numerical methods to numerically solve differential and integral equations. I think mostly of the analytical aspects. > I bet that you can randomly choose any fundamental theorem of > linear algebra and the chances are big that you can use them somehow > in solving calculus problems. Most of the time, the fundamental theorems > are used to prove more advanced theorems which are then subsequently > used in other fields (such as calculus). The Rank-Nullity theorem is > such a fundamental theorem that is used in proving many other results. When you write that many theorems can be used to solve calculus problems, do you have in mind the solution of specific equations or properties of many calculus functions? That is, would you use theorems from linear algebra to solve dy/dx = x^2 + y^2, if they were applicable. Such as eigen vectors, numbers? I suppose it would be useful and elegant to apply such theorems, but I, personally, aren't familiar at this point with the theorems to use them in calculus. It would be nice to though. === Subject: Re: Vector spaces and calculus > When you write that many theorems can be used to solve calculus >problems, do you have in mind the solution of specific equations or >properties of many calculus functions? That is, would you use theorems >from linear algebra to solve dy/dx = x^2 + y^2, if they were applicable. >Such as eigen vectors, numbers? Yes, you are very close! For example, the task of solving the differential equation dy/dx = k y is identical to the task of finding an eigenvector (eigenfunction) for the linear map D : V --> V , where D = d/dx is the derivative operator and V is the vector space of functions you are interested in (e.g. the space of smooth functions which satisfy some boundary conditions). Someone mentioned Functional Analysis, which is related to Harmonic Analysis, which is related to harmony: attempts to describe the motion of a plucked string whose ends are pinned down require finding the eigenvectors of a mapping like L = D^2 . (The eigenvectors are the spans of functions like f(x) = sin( 2 pi n x ) or cos( 2 pi n x ), where n is constrained to be integral by definition of V -- a consequence of the string ends being fixed. To tie in with another thread, let me note that you verify these functions are eigenvectors by noting that L(f) is a multiple of f, not by computing a determinant.) dave === Subject: Re: Vector spaces and calculus >The use of linear algebra breaks down a bit when we start talking about >infinite dimensional vector spaces, and most of the time differential >equations involve functions that are elements of such spaces. Linear algebra as it's usually done is just a special case of functional analysis where the spaces are finite dimensional. Of course you can also do linear algebra with all sorts of non-fields like boolean algebras. === Subject: nested closed set theorem for R^n Is there an easy was to prove the nested closed set theorem for R^n using the Bolzano Weierstrass theorem? I've seen a proof that doesn't use it, but my book says to use it as a Hint, so I'm wondering if it comes out easier. However, I do not see how to apply it to the situation. === Subject: Re: nested closed set theorem for R^n Can't you just pick a point out of each set ,the resulting sequence should have a convergent subsequence and its limit should be the required point.Justify all this using your hypothesis and the Bolzano === Subject: Re: Euclid & Aristotle never had the concept of Mathematical Consistency, so did they have Reductio Ad Absurdum > Aristotle used Reductio Ad Aburdum in his Prior Analytics to reduce the > syllogisms Baroco (second figure) and Bocardo (third figure) to the perfect > syllogism Barbara. David, I think it is fair to say that when Reductio Ad Absurdum is discovered as a method of proving that there is some degree of conversation about it and not as a jigsaw puzzle piece in a large classification scheme. When Reductio ad Absurdum was discovered in history then there was some lengthy discussion about it and then also there was some display of how it is useful as a technique. And some discussion as to how and why it works. Clearly, Aristotle did not do that for Reductio Ad Absurdum. The below quotes indicate that Aristotle may have had a Classification scheme of various argument forms where Reduce to Impossibility may have faint glimmers to Reductio Ad Absurdum. And Smith's is a mere interpretation or translation. The discoverer and discovery of Reductio Ad Absurdum should be free of interpretation. Should be so clear that there is little doubt Reductio is what it is and how to use it. Let us play a game with Quantum Mechanics that the mathematicians have played with Reductio Ad Absurdum. Suppose the writings of Democritus still existed or suppose the writings of John Dalton were analyzed and suppose with the Atomic theory of Democritus or Dalton the words comes in lumps or the word discrete is found in those writings. Are we then to make claim and to fill the history of physics textbooks saying that Quantum Mechanics was discovered by Democritus in Ancient Greek time or discovered by Dalton before Planck ever had his constant? Of course not. Just because we find a few words of comes in lumps or discrete in Democritus or in Dalton does not mean that these men discovered Quantum Mechanics. Likewise for Aristotle, Pythagoras, Euclid, Archimedes. Just because they used the terms of reduce to impossible or per impossible does not mean they discovered Reductio Ad Absurdum. If Aristotle discovered Reductio Ad Absurdum then there should be a clear and lengthy discussion about it and clear use of the method. Above and beyond translation or interpretation by editors or readers or translators. Neither Aristotle or Euclid discuss Reductio Ad Absurdum in length. Nor do they show any use of it in a clear, unmistakable case. In every instance of a claim that Aristotle or Euclid or Pythagoras or Archimedes used Reductio Ad Absurdum, is a case of how the translators are translating the Ancient Greek manuscripts. --- quoting from http://humanities.byu.edu/philosophy/aporia/volumes/vol111/christensento.htm l 5. Conversion is Aristotle.89s usual method of proof, by which he uses the first-figure syllogisms as paradigm cases. .8bImpossibility.8a is the word Smith uses to translate reductio ad absurdum. --- end quoting from http://humanities.byu.edu/philosophy/aporia/volumes/vol111/christensento.htm l --- quoting from http://planetmath.org/encyclopedia/AristotelianLogic.html If a `c' follows an `o' it indicates that the proof must be done reductio ad impossibile.(There are two such syllogisms, Bocardo and Baroco which are not correctly proven by Aristotle. To see the reason why 17.) If an `s'(or `p') follows the first or second vowel, this indicates that the proposition corresponding to this vowel is simply(or per accidens) converted in the proof. If an `s'(or `p') follows the last vowel this implies that the conclusion will be derived by making a simple(or per accidens) conversion of the conclusion of the first figure syllogism that is used in the reduction. Finally an `m' indicates a rearrangement of the premises to satisfy the order: major premise, minor premise and conclusion. --- end quoting from http://planetmath.org/encyclopedia/AristotelianLogic.html To say that Aristotle knew of Reductio Ad Absurdum, is to say that Dalton, when and if he had written the word lump or discrete knew of and discovered Quantum Mechanics. Absurd! Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Philo/Aristotle on consistency Re: Euclid & Aristotle never had the concept of Mathematical Consistency, so did they have Reductio Ad Absurdum > suppose the writings of John Dalton were analyzed and suppose with the Atomic > theory of Democritus or Dalton the words comes in lumps or the word > discrete is found in those writings. Are we then to make claim and to fill > the history of physics textbooks saying that Quantum Mechanics was discovered > by Democritus in Ancient Greek time or discovered by Dalton before Planck ever > had his constant? Of course not. Just because we find a few words of comes in > lumps or discrete in Democritus or in Dalton does not mean that these men > discovered Quantum Mechanics. I dare not search under Google for consistency of Euclid for fear that the first 10,000 hits will be of the modern day parallel postulate. So I searched for Aristotle on consistency and found Philo: --- quoting http://plato.stanford.edu/entries/dialectical-school/ So according to Boethius the basic feature of Philonian modalities is some intrinsic capability of the propositions to be or not to be true or false. That this feature is intrinsic is plain from the phrases ëby its own nature.89 and ëin itself.89. In one source, both phrases are used to characterize Philonian possibility (Simplicius, On Aristotle's Categories 195); hence both phrases may have originally applied to all four accounts. In all sources the concept of possibility stands out, and so it seems likely that Philo built his set of modal notions on a concept of internal consistency, as given in his account of possibility. Philo's modal concepts are thus defined by resort to another, perhaps more basic, modal concept. As to the kind of consistency Philo had in mind, we learn nothing more. Notwithstanding this, there can be little doubt that Philo's modal concepts satisfy a number of basic requirements which normal systems of modern modal logic tend to satisfy, too. (I assume here that Philo accepted the principle of bivalence, as we have no contrary information.) These requirements are: 1. Every necessary proposition is true and every true proposition possible; every impossible proposition is false and every false proposition non-necessary. --- end quoting http://plato.stanford.edu/entries/dialectical-school/ You see, Quantum Mechanics cannot start without a developed Planck's constant. Anything else would be shadowy intellectualizing or guessing that amounts to little. So is there a contributing advance factor that one can say that Reductio Ad Absurdum could not exist in a time and place without this other factor? Analogy: we could not have the Maxwell Equations by Maxwell if Coulombs law and Faradays law and Ampere's law had not existed. Those were contributing factors that made the time ripe and ready for Maxwell to do the synthesis. So let us look at Reductio Ad Absurdum and what minimum background factors need be existing in order for Reductio Ad Absurdum to exist as a method. I would say the minimum requirement is a full blown theory of consistency. Because Reductio Ad Absurdum is a tool, a part and parcel of consistency. To be consciously aware of Reductio Ad Absurdum cannot take place in an environment that is primitive to the concept of consistency. So, in the above quoting and a little further reading the state of the art of the concept of Consistency in Ancient Greek times during Euclid, Aristotle and even Archimedes, consistency was primitive. Too primitive to give birth to Reductio Ad Absurdum. If Reductio Ad Absurdum had existed in Ancient Greek times then their development of consistency would have been extensive. So that leaves the awkward question of when and where was consistency developed to the point where Reductio Ad Absurdum was borne and developed. Archimedes Plutonium www.iw.net/~a_plutonium whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies === Subject: Re: Philo/Aristotle on consistency Re: Euclid & Aristotle never had the concept of Mathematical Consistency, so did they have Reductio Ad Absurdum <42707E77.C26F9A1D@iw.net> <42712996.C3512D36@iw.net> by this Philonian account, impossibility implying falsity implying no neccesity for the proposition, the possibility of contradiction is established; is it not? >b) the hare-brained analysis of WTC7 assumes that the substructure of the complex, which included a large subway station, was not affected by the two largest buildings, falling down into it. --Chair Man George NAND Strep Throat @ W'gate? http://tarpley.net/bush12.htm http://laroucehpub.com http://members.tripod.com/~ame[CapitalEth]rican almanac === Subject: Re: SF: Abstracted SFT, issue of triviality <426A7B4E.B08E21B7@ix.netcom.com> <87br80sskz.fsf@phiwumbda.org> Discussion, linux) > Then again, the discussion is over infinity, so necessarily, any > pattern imaginable can occur[...] >>Whoa. That's deep. >>You know lots of stuff about infinity, I bet. > If you can't spell use a friggin spell checker! > (The word is lotsa...) Lots is a word. So is of. Spell checkers don't help with that kind of mistake. But I promise to profread my posts good in the future. -- If you people knew really, in your heats [sic], and minds, who I actually am, would you even reply to my posts? I'd probably get that hero worship crap. -- JSH explains why the greatest mathematician in the world masquerades as a moron. === Subject: Re: SF: Abstracted SFT, issue of triviality >> Then again, the discussion is over infinity, so necessarily, any >> pattern imaginable can occur[...] >Whoa. That's deep. >You know lots of stuff about infinity, I bet. >> If you can't spell use a friggin spell checker! >> (The word is lotsa...) >Lots is a word. So is of. Spell checkers don't help with that >kind of mistake. Good point, sorry. >But I promise to profread my posts good in the future. ************************ David C. Ullrich === Subject: Re: SF: Abstracted SFT, issue of triviality <426A7B4E.B08E21B7@ix.netcom.com> <87br80sskz.fsf@phiwumbda.org> <87mzrjjusf.fsf@phiwumbda.org> <2ae171d6lvhgt44qf66oknnu5ahol3ll1e@4ax.com> (The word is lotsa...) >Lots is a word. So is of. Spell checkers don't help with that >kind of mistake. > Good point, sorry. >But I promise to profread my posts good in the future. But the grammar checker should have caught it... Iain === Subject: Re: SF: Abstracted SFT, issue of triviality <426A7B4E.B08E21B7@ix.netcom.com> <87br80sskz.fsf@phiwumbda.org> <87mzrjjusf.fsf@phiwumbda.org> <2ae171d6lvhgt44qf66oknnu5ahol3ll1e@4ax.com> Discussion, linux) > (The word is lotsa...) >>Lots is a word. So is of. Spell checkers don't help with that >>kind of mistake. >> Good point, sorry. >>But I promise to profread my posts good in the future. > But the grammar checker should have caught it... Linux users don't need no stinkin' grammar checker. -- I'm the guy. I have always been the guy. Your post will sit here for a while, soon be ignored, except for people coming to read my reply, and your satisfaction will fade as you move on, and I'll still be the guy. -- James S. Harris will *always* be the guy. Duh. === Subject: Re: SF: Abstracted SFT, issue of triviality !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw >> (The word is lotsa...) Lots is a word. So is of. Spell checkers don't help with that >kind of mistake. > Good point, sorry. >But I promise to profread my posts good in the future. >> But the grammar checker should have caught it... > Linux users don't need no stinkin' grammar checker. Like offering a guitar tuner to Keith Richards, eh? -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: SF: Abstracted SFT, issue of triviality <426A7B4E.B08E21B7@ix.netcom.com> <87br80sskz.fsf@phiwumbda.org> <87mzrjjusf.fsf@phiwumbda.org> <2ae171d6lvhgt44qf66oknnu5ahol3ll1e@4ax.com> <87pswf10dh.fsf@phiwumbda.org> <85vf6757fd.fsf@lola.goethe.zz> lots o'what? >b) the hare-brained analysis of WTC7 assumes that the substructure of the complex, which included a large subway station, was not affected by the two largest buildings, falling down into it. --Chair Man George NAND Strep Throat @ W'gate? http://tarpley.net/bush12.htm http://laroucehpub.com http://members.tripod.com/~ame[CapitalEth]rican almanac === Subject: Re: Groups of order 36 [...] > To find the other > non-abelian group of order 36 with no normal subgroup of order 9 > you need to look for an action of Z_9 on Z_2 x Z_2. What n.-t. > h. is there from Z_9 to Aut(Z_2 x Z_2)? > It must divide 9, so is the element of order 3. > G = (Z_2 x Z_2) x| Z_9 where the semi-direct product is given by this > element of order 3. Yes. Specifically, it has a presentation . Note that acts trivially on , per its being the kernel of the relevant homomorphism Z_9 -> Aut(Z_2 x Z_2). -- Jim Heckman === Subject: Re: Richard Henry shows his psychopathology > Lies, lies, and more lies. > You got that wrong. It's lies, damned lies and statistics :) Hey Giuseppe. Fancy meeting you down here in the bowels of usenet depravity. The whole Berlusconi 'confidence vote' thing got you depressed? Whatever you do, don't kill yourself in rsa, bunch of cold-blooded non-caring bastards here, they'll only laugh, steal your shoes and then argue about how to properly bind them to a splitboard. === Subject: Re: Richard Henry shows his psychopathology > Lies, lies, and more lies. >> You got that wrong. It's lies, damned lies and statistics :) > Hey Giuseppe. Fancy meeting you down here in the bowels of usenet > depravity. The whole Berlusconi 'confidence vote' thing got you > depressed? Whatever you do, don't kill yourself in rsa, bunch of > cold-blooded non-caring bastards here, they'll only laugh, steal your > shoes and then argue about how to properly bind them to a splitboard. I don't ski. At least not when in bed with flu. -- Giuseppe Oblomov Bilotta Axiom I of the Giuseppe Bilotta theory of IT: Anything is better than MS === Subject: Re: Richard Henry shows his psychopathology > I don't ski. At least not when in bed with flu. Skiing is GREAT for flu. as ski instructors in the US don't get sick leave, I have learned to just keep on working. And oddly, my flus aren't as serious as they seemed when I was in bed for weeks in Australia (with 2 weeks full pay/2 weeks half pay sick leave accruing each year). Mind you, flu drugs have improved too. ant === Subject: Re: Richard Henry shows his psychopathology > Skiing is GREAT for flu. as ski instructors in the US don't get sick leave, > I have learned to just keep on working. And oddly, my flus aren't as serious > as they seemed when I was in bed for weeks in Australia (with 2 weeks full > pay/2 weeks half pay sick leave accruing each year). Mind you, flu drugs > have improved too. Well, considering how even walking from the bed to the toilet and back is staggering, I suspect I wouldn't be able to ski when fluent (ehehehe). -- Giuseppe Oblomov Bilotta Axiom I of the Giuseppe Bilotta theory of IT: Anything is better than MS === Subject: Re: Richard Henry shows his psychopathology >> Skiing is GREAT for flu. as ski instructors in the US don't get sick >> leave, I have learned to just keep on working. And oddly, my flus >> aren't as serious as they seemed when I was in bed for weeks in >> Australia (with 2 weeks full pay/2 weeks half pay sick leave >> accruing each year). Mind you, flu drugs have improved too. > Well, considering how even walking from the bed to the toilet and back > is staggering, I suspect I wouldn't be able to ski when fluent > (ehehehe). that's how I used to be. I'd get delirious and it was awful. This season I did go to bed, as I had a raging fever, but I did turn up for 9am lineup, and then shuffled home to bed. And on the 4th day, I went for a ski after lineup, and improved from them on. It's scary what you can do when technically very sick. Having said that, I think I prefer a country where you get sick leave. ant === Subject: Re: Surrogate factoring, mapping, hyperbolas Discussion, linux) > I always win, as I have from the beginning. Posters lie about that > reality and go on for a while, but you seem to be a little more > intelligent, and not interested in merely being a lackey to JSH. You always win, but somehow you seem to still have no positive recognition for your accomplishments. It's a fairly muted sense of winning, ain't it? -- Jesse F. Hughes [I]t's the damndest thing. There's something wrong with every last one of you, and I *never* thought that was a possibility. But now I feel it's the only reasonable conclusion. --JSH sees some sorta light === Subject: Finite Groups Book Hi all: I was told that after D. Gorenstein passed away, one can find his books (to download, I mean) in the web. And indeed I found his books finite simple groups I,II,III, etc., but I can't find his book called plainly Finite groups (it is from 1968, but there was a 2nd edition in 1980). Does any of you guys know where on the web can I find it? I'd very much appreciate anybody's help with this. Tonio === Subject: Re: Factoring problem and the SFT jstevh@msn.com says... > But I need help. > Someone needs to contact a government agency, say that hey, they need > to pay attention to this, and I can go and give a complete briefing. Okay, it's been done for you since you don't want to do it yourself. > That's the trouble with examples. Yeah, they do tend to expose flaws in improperly formed theorems, or more often, pseudo-theorems. How inconvenient. > I need to brief someone in government. I agree. They need to be aware of your tremendous accomplishments. > Now if you acknowledge that my research MIGHT be important, then you > can agree with me that it needs to be taken to areas off Usenet where > some serious research can take place behind closed doors. I definitely agree that it needs to be taken off usenet, and I don't even agree that it might be important to anyone but you. > I'd prefer the NSA. Good. You have been booked to present to the NSA's Domestic Technology Transfer Program steering committee, at 8:30am on the morning of Thursday, June 2nd, 2005. The meeting will be held in the office at 9800 Savage Road, Suite 6541, Fort Meade, MD 20755-6541. When you arrive, just tell them your name and that you are The Usenet guy that invented the SFT, they'll be expecting you. Do not be alarmed, the needle won't hurt for long. I'm hoping a subscriber to one of these groups that lives in the area will volunteer to photograph your arrival for posterity. We will all be so very proud of you. Good luck, and you better start working on those powerpoint charts ASAP. -- Randy Howard (2reply remove FOOBAR) Making it hard to do stupid things often makes it hard to do smart ones too. -- Andrew Koenig === Subject: Re: Factoring problem and the SFT <4NqdnQkT9v0JRvLfRVn-pA@hamilton.edu> yeah, a seminar! >b) the hare-brained analysis of WTC7 assumes that the substructure of the complex, which included a large subway station, was not affected by the two largest buildings, falling down into it. --Chair Man George NAND Strep Throat @ W'gate? http://tarpley.net/bush12.htm http://laroucehpub.com http://members.tripod.com/~ame[CapitalEth]rican almanac === Subject: Re: Factoring problem and the SFT Discussion, linux) > In answer posters have replied with a deluge of pseudo-math, which may > be sufficient for policymakers, cryptologists with an investment in RSA > and people who just want to sleep at night, but is it ok just to be > comforted against the facts? > The reality is there is no mathematical reason to believe that the SFT > is picky. > BUT with posters comforting people who are supposed to worry about such > things with the idea that if it were possible it might work so many > people wouldn't argue against it, that just leaves the people willing > to check, capable of exploiting the mathematics in secret. Do you really think that policy makers are concerned by your posts on SFT but then comforted by the posts of your respondents? You think that cryptologists turn to sci.math (or sci.crypt) to check whether the jig is up? Really? -- Jesse F. Hughes [I]f gravel cannot make itself into an animal in a year, how could it do it in a million years? The animal would be dead before it got alive. --The Creation Evolution Encyclopedia === Subject: Re: Factoring problem and the SFT >The SFT provides a solution to >sqrt(x^2 - 4A^2(A^2 - B^2)y^2) >with all non-zero integer, where x/y is determined by the rational >factorization of B^2(A^2 - B^2). >That gives you what the SFT does in a nutshell. >And that reality is not in doubt. The SFT does in fact give a > solution >to the square root shown. >>That's not in doubt, hasn't been for a couple of weeks. > But that has never before been achieved in mathematical history. Perhaps, but having a new result has little to do with the significance of the result. In essence, all the SFT (new version) says is If g|B^2(A^2 - B^2) and f_2 = g - (A^2 - B^2), then there is a rational number f_1 such that f_1 f_2 = A^2(A^2 - B^2). But that's ALWAYS true, regardless of how f is defined: Using your quadratics, using my simplification of your quadratics, using any other formula for f_2, no matter how simple or complicated, it will ALWAYS be true as long as f_2 != 0. Seems pretty useless to me. that while it may appear to be picky over some finite run, there is > just a probability for that, and some simple reasons why it might occur > at particular places, which I've already worked out! So you claim. In fact, you haven't, as numerous posters have pointed out. can agree with me that it needs to be taken to areas off Usenet where > some serious research can take place behind closed doors. That's a great idea, especially the part about taking if off Usenet. > I'd prefer the NSA. Go for it. Rick === Subject: Re: Factoring problem and the SFT <440071pbcgvussqql4vns1c6ej2mi373j9@4ax.com On 27 Apr 2005 13:51:46 -0700, Proginoskes > [snip, body part removed] > While SFT is not verified to work cut sci.crypt from replies. SFT's > current state belongs to other groups than sci.crypt. > Juuso what is it with you and your obsession with sci.crypt? do you feel you own it or you are a big shot over there? if you are bothered by James' posts then either ignore it or killfile him for god sake. stop whining on sci.math about how precious sci.crypt is being soiled by posts form JSH. it sounds really silly okay! === Subject: Re: Factoring problem and the SFT [Rick Decker] [...] > Now if we knew A = p * q was the product of two distinct > primes and if we had no reason to assume anything about > some set of N numbers, then we would expect N/p of them > to be divisible by p, N/q to be divisible by q, and > N/pq to be divisible by both p and q. So we'd expect > N/p + N/q - N/pq > useful factors. Sheesh: you and Nora _both_ lying to James about this, week after week. No wonder he calls Nora a liar and you lying scum . Each integer divisible by both p and q was counted once in N/p, and again in N/q, so you need to subtract 2N/pq to get the count of integers divisible by p or q but not both. The result holds exactly then if the universe from which we choose consists of a contiguous range of exactly iN integers (for some non-negative integer i). Note that this favors James, because it lowers the chance of winning by luck (i.e., makes random-gcd easier to beat). The difference becomes negligible as min(p, q) increases, but all SFT-ish algorithms to date have done so poorly that nobody ever gets to RSA-sized p and q before giving up (e.g., in the last round of this stuff, I never went beyond testing products of pairs of 20-bit primes, and toward the end cut that to 15-bit primes as the algorithms got ever more expensive to try). BTW, I'd be happy to test SFT with that framework too, except James has never given an actual algorithm for this one. I'm not going try to picking arbitrary rationals out of thin air at the end. If looking (just) at all integer factors of B^2(A^2 - B^2) would satisfy James, that would be non-insane enough to test. > ... If this held in general, it would mean that SFT would yield useful > results at a rate NO BETTER THAN RANDOM CHOICE. > I've tried enough examples to be mostly convinced > that's just what happens with your theorem. Having tested some dozen of these before, and seeing nothing essentially new in this one, I'm almost entirely sure of that outcome. > If it is, then you're in deep doo-doo when you attempt > to factor a big A: if A = pq and p and q are hundred-digit > primes, that would give you about a 2 in 10^100 chance > that one of your choices of g_1 and g_2 would lead > to a useful factorization of A. I wouldn't hold my breath. Exactly so, but JSH can't believe it. He believes in the math, and math doesn't lie -- which in this case is an absurd argument about infinite sets: The set of all integers divisible by 101 is (countably) infinite. The set of all integers not divisible by 101 is (countably) infinite. Therefore half of all integers are divisible by 101. I don't believe he understands enough set theory to recognize that this _is_ his argument, so with a bit of luck I'll rate lying scum too for pointing out that it is (he doesn't appear to know that there's a bijection between _any_ two countably infinite sets, and spaced out when that was explained before). === Subject: Re: Factoring problem and the SFT > [Rick Decker] > [...] N/q, so you need to subtract 2N/pq to get the count of integers divisible by > p or q but not both. The result holds exactly then if the universe from > which we choose consists of a contiguous range of exactly iN integers (for > some non-negative integer i). Heh. Yup, you're right. never given an actual algorithm for this one. I'm not going try to picking > arbitrary rationals out of thin air at the end. If looking (just) at all > integer factors of B^2(A^2 - B^2) would satisfy James, that would be > non-insane enough to test. That's all you need. Including rational factors just adds a random choice on top of something that already works no better than random choice. You're right, though, that this latest incarnation of SFT doesn't qualify as an algorithm since it makes no mention of how the rational factors are to be selected. Lacking that, it's just as trivial as the previous versions. Any nonzero rational divides any other. Rick === Subject: Re: Factoring problem and the SFT !3KEIp?*w`|bL5qr,H)LFO6Q=qx~iH4DN;i;/yuIsqbLLCh/!U#X[S~(5eZ41to5f%E@'ELIi $t^ VcLWP@J5p^rst0+('>Er0=^1{]M9!p?&:z]|;&=NP3AhB!B_bi^]Pfkw > That's all you need. Including rational factors just adds a > random choice on top of something that already works no better > than random choice. You're right, though, that this latest > incarnation of SFT doesn't qualify as an algorithm since it > makes no mention of how the rational factors are to be selected. > Lacking that, it's just as trivial as the previous versions. > Any nonzero rational divides any other. Well, but is the result properly a unit? -- David Kastrup, Kriemhildstr. 15, 44793 Bochum === Subject: Re: Factoring problem and the SFT <4NqdnQkT9v0JRvLfRVn-pA@hamilton.edu> [...] > Now if we knew A = p * q was the product of two distinct > primes and if we had no reason to assume anything about > some set of N numbers, then we would expect N/p of them > to be divisible by p, N/q to be divisible by q, and > N/pq to be divisible by both p and q. So we'd expect > N/p + N/q - N/pq > useful factors. > Sheesh: you and Nora _both_ lying to James about this, week after week. No > wonder he calls Nora a liar and you lying scum . They're lying to the public. Their posts are meant to influence others, not me. Now I can now prove that the SFT has to work. It's not even complicated. If you consider f_1 = (-(z - 2A^2)+ sqrt((z - 2A^2)^2 - 4A^2(A^2 - B^2)))/2 where z is an integer, you have a finite set of solutions, where at least one value of z must factor A non-trivially. Well, it turns out that you can solve the equations that define z, for rational factorizations of B^2(A^2 - B^2) such that you get an integer z. It's not even hard to do the math. James Harris === Subject: Re: Factoring problem and the SFT <4NqdnQkT9v0JRvLfRVn-pA@hamilton.edu> Now I can now prove that the SFT has to work. Huzzah! > It's not even complicated. Huzzah! Huzzah! > If you consider > f_1 = (-(z - 2A^2)+ sqrt((z - 2A^2)^2 - 4A^2(A^2 - B^2)))/2 > where z is an integer, you have a finite set of > solutions, Huzzah! Hu- Er ... last time I checked, the set of integers was _infinite_. Would you care to put a bound on z, or on the number of integers you have to consider? > where at least one value of z must factor A non-trivially. Well, if A is composite, then at least one value in the set {2, 3, ..., floor(sqrt(A))} factors A non-trivially. > Well, it turns out that you can solve the equations that define z, for > rational factorizations of B^2(A^2 - B^2) such that you get an integer > z. > It's not even hard to do the math. So do it, and give us an algorithm already! --- Christopher Heckman === Subject: Re: Factoring problem and the SFT <4NqdnQkT9v0JRvLfRVn-pA@hamilton.edu> Now I can now prove that the SFT has to work. > Huzzah! > It's not even complicated. > Huzzah! Huzzah! > If you consider > f_1 = (-(z - 2A^2)+ sqrt((z - 2A^2)^2 - 4A^2(A^2 - B^2)))/2 > where z is an integer, you have a finite set of > solutions, > Huzzah! Hu- > Er ... last time I checked, the set of integers was _infinite_. > Would you care to put a bound on z, or on the number of integers you > have to consider? There are only a finite number of integer z's such that sqrt((z - 2A^2)^2 - 4A^2(A^2 - B^2)) is an integer. It's fairly easy to show how to solve for integer z's. That shows just one way to guarantee a factorization using equations from the SFT. James Harris === Subject: Re: Factoring problem and the SFT <4NqdnQkT9v0JRvLfRVn-pA@hamilton.edu [...] > I have the solution for > sqrt(x^2 - 4A^2(A^2 - B^2)y^2) It's impossible to solve an expression, by definition. You can solve equations, or inequalities, or anything with a verb, but not expressions. Expressions no verb. You can simplify expressions or evaluate expressions, though. > [...] > But I need help. When you post statements like this, you open up yourself to ALL SORTS of criticism. > Someone needs to contact a government agency, say > that hey, they need to pay attention to this, and > I can go and give a complete briefing. Why don't you do it? You're the best qualified, because you've spent the most research on the SFT, and you've managed to produce an algorithm when no one else has. Oh, wait, I know. You haven't even shown how to factor a single number concretely. And I'm sure one of the first things the NSA will say is: What numbers have you factored with this method? > That's the trouble with examples. > that's just what happens with your theorem. If it > is, then you're in deep doo-doo when you attempt > to factor a big A: if A = pq and p and q are > hundred-digit primes, that would give you about a > 2 in 10^100 chance that one of your choices of > g_1 and g_2 would lead to a useful factorization > of A. I wouldn't hold my breath. > Examples do not prove. But they can DIS-prove. For instance, if I say I have the following theorem: THEOREM. For all integers n, 1 + 3 n is a perfect square. This can be disproven by a single example: What happens when n = 3? Then 1 + 3 n = 10, which is not a perfect square, so the theorem is false, no matter how much I believe it's true (it works for n = 0, n = 1, n = 5, etc.). I sense that's the true reason why you fear examples. Examples have the power to negate decades of research, but cannot support it. So it is in your best interest to NOT provide examples, or to even look at them, because that would convince you that SF is worthless, and you simply don't want to face that option. > I can prove that the SFT cannot be picky over infinity, > which means that while it may appear to be picky over > some finite run, there is just a probability for that, > and some simple reasons why it might occur at particular > places, which I've already worked out! What are your particular places for when M = 12,321? > I need to brief someone in government. > Now difference of squares is well enough known that > the more rational among you who haven't been arguing > with me for years now calling me nasty names like > crank can see that the research MIGHT be important. If the name fits, use it. > Now if you acknowledge that my research MIGHT be > important, But not in the way you're using it. > then you can agree with me that it needs to be taken > to areas off Usenet where some serious research can > take place behind closed doors. Or you could simply stop posting about it. The more you post, the more that people have to work with. > I'd prefer the NSA. Try it, dude. --- Christopher Heckman === Subject: Re: Factoring problem and the SFT <4NqdnQkT9v0JRvLfRVn-pA@hamilton.edu I need to brief someone in government. all crackpots feel they need to brief some big guys... you should chat with the crackpot George Hammond on sci.physics and have a little heart-to-heart. > Now if you acknowledge that my research MIGHT be important, then you > can agree with me that it needs to be taken to areas off Usenet where > some serious research can take place behind closed doors. > I'd prefer the NSA. hua ha ha ha, i always find this very funny... can't stop laughing because of you, maths crank. silly crackpot. === Subject: Re: Factoring problem and the SFT <440071pbcgvussqql4vns1c6ej2mi373j9@4ax.com On 27 Apr 2005 13:51:46 -0700, Proginoskes > [snip, body part removed] > While SFT is not verified to work cut sci.crypt from > replies. SFT's current state belongs to other groups > than sci.crypt. My post didn't have anything to do with the SFT. It had to do with the expected requirements for the NSA to take James Harris sincerely. --- Christopher Heckman P.S. Can I talk about spiral colorings? === Subject: Re: Factoring problem and the SFT <4NqdnQkT9v0JRvLfRVn-pA@hamilton.edu> can agree with me that it needs to be taken to areas off Usenet where >some serious research can take place behind closed doors. >I'd prefer the NSA. > Your post come accidently even to sci.crypt. Cryptography can Nope. It wasn't an accident. James Harris === Subject: Re: One-point compactification of Q 1st countable Hausdorff space is KC. > This is actually a watered-down version of the following theorem: > The one-point compactification of a space is KC > if and only if the original space is a KC k-space. > The other compact sets of S_p are compact sets containing p. > Why are these sets closed? > Let K be such a compact set. > K compact subset S_p containing p? > Suppose it's not closed, so there's a point c in cl(K)K. > c /= p > There's a sequence x_n in K/S converging to c > (as S is 1st countable). > c has a countable local base all of whose members are in S. > And since c is in cl(K) we can use this to construct a sequence > in K converging to c. And this sequence will also be in S. Not necessarily, however only finitely many of it's tails won't be. > Let L = { x_n | n in N } / {c}. > Then L is a compact subset of S, so it's closed in S_p. > Therefore L/K is compact, and so closed in S_p. > No, what I meant is that L is closed (in S_p) > and K is compact, so L/K is compact. Ok, skip the and so closed in S_p. > But that's impossible, as (x_n) is a sequence in L/K > that converges to a point outside of L/K. Where is Hausdorff used? As L/K is compact subset S and S is 1st countable, so is L/K. Hence L/K sequentially compact and (x_n) has subsequence converging to point in L/K. Now the subsequence also converges to same point in S and in addition since (x_n) -> c within S, so does the subsequence. As S is Hausdorff, c not in L and other point in L/K are the same, ouch! ---- === Subject: Re: start me off (again) #3b you have enlightened me === Subject: Your account #3M7888 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) with ESMTP id j3S8wPh29531; by support2.mathforum.org (8.12.10/8.12.10/The Math Forum, $Revision: 1.6 secondary) with SMTP id j3S8wgjZ020870; by silk.eke.pochta.ru (Cyrus v2.2.3) with LMTPA; post-comp-soft-sys-matlab@mathforum.org, post-sci-math@mathforum.org, pow@mathforum.org, pow-teach@mathforum.org, r-fathom@mathforum.org, richard@mathforum.org, roya@mathforum.org, ruth@mathforum.org, rxp8bgtqv5xp@mathforum.org, salejan@mathforum.org, sarah@mathforum.org We tried contacting you awhile ago about your low interest morta(ge rate. You have qualified for the lowest rate in years... You could get over $380,000 for as little as $500 a month! Ba(d credit? Doesn't matter, low rates are fixed no matter what! To get a free, no obli,gation consultation click below: http://www.crazy-biz.net/sign.asp Carl Madison to be remov(ed: http://www.crazy-biz.net/gone.asp this process takes one week, so please be patient. we do our best to take your email/s off but you have to fill out a rem/ove or else you will continue to recieve email/s. === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability > If one presents a formalism but doesn't show that it represents a > particular logical argument, then they have not shown that the > formalism has any mathematical significance. The paper about which you are having difficulties is a -system description- paper, or more specifically, the description of how the system works in one specific input. It's intent is to convince the reader that their system can do what it claims to do. An analogous paper might be a paper about a new process scheduling algorithm that does not seek to prove the algorithm correct but rather to show efficiency comparisons with other algorithms. So yes there is a matter of trust that the authors ran their system in an appropraite way, saving the reader the effort of having to download the software and do the same experiments and analyze the copious output. This is in great contrast to a mathematical paper, one which attempts to convince the reader with -informal- proof of the truth of the claim, mostly in symbolic terms. Boyer and Moore were not attempting in this paper to -prove- that their system could prove the undecidability of the halting problem (UHP). They did not present a mathematical proof of UHP (that is considered to be basic knowledge of the expected readership). They didn't even present the text of the output of their system when given a suitably symbolized version of UHP (they gave parts of the encoding in the paper). All they did was give pieces of these different symbolizations, and some explanation of how the internals of their system was supposed to deal with this special example. It is probably confusing that the system they are presenting is a theorem-proving system. Here's a hopefully not too simple analogy: suppose some research group creates a new articulated robot. They publish all sorts of papers describing its properties: how to operate it, obstacle avoidance algorithms, how it was built, etc. They write a couple of books, even make the plans available publicly. And, by the way, they also write a short paper on how you can write a program to make the robot climb stairs. No pictures, just some descriptions, diagrams, explanation that the center of mass is not too high when the front feet are up on the next step, etc. Also, by the way, after 20 some years a robot based on the same plans is used to weld car parts together (and dance in music videos). So basically you doubt that they ever got their robot to walk upstairs. Skepticism is fine but the rest of the robot..ahem... theorem proving community pretty much accepted it long ago. The paper in question convinced them, not by proving UHP (everybody knows that's true), not by printing out the entire Lisp code of the prover (too long), or by proving that their prover is correct (too laborious and justified elsewhere (in their books)), or by giving the entire input for describing UHP (they gave a lot of it but all of it would be too long) or by giving the output/the proof (way, way too long), but by giving enough telling details that a reader (in the ATP community), by filling in the details, would feel confident that the authors established their claim. Their paper does not give a -mathematical- proof, either in use or mention. On looking at the paper very closely, I agree with you in principle. You're right, there's no proof of -any- kind in that paper (well, no theorems, and so no proofs needed to back them up). But that paper (and what else I know about the Boyer-Moore prover) certainly convinces me that they have done it, been able to prove it automatically (with input describing the HP situation). I vaguely recall that you claim to have written a theorem proving system that walks up...sorry... proves UHP. Excellent!! As cool as that is, it is old news. Maybe you did it in a new and general way. Maybe you did it in 5 lines of ML code (and the same 5 lines will prove the 4 color theorem with only 10 cases). To impress the theorem proving crowd nowadays you'll probably have to convince them that your system does the proof in a particularly special way. Oh.. to your original point.. > If one presents a formalism but doesn't show that it represents a > particular logical argument, then they have not shown that the > formalism has any mathematical significance. Yes, I agree, in some very general sense. But the paper in question is definitely -not- self-contained. One would need to be acquainted with the rest of their work to be convinced that their formalism has mathematical significance. -- Mitch Harris (remove q to reply) === Subject: Re: A Simple Question on Boyer & Moore's Mechanical Proof of Halting Unsolvability <3dbn4rF6o177mU1@news.dfncis.de The paper is a > -system description- paper, or more specifically, the > description of how the system works in one specific input. But it gives no output from that system. > It's intent is to convince the reader that their system > can do what it claims to do. Then why not show the proof? Wouldn't that go a long way toward that goal? > So yes there is a matter of trust that the authors ran > their system in an appropraite way, saving the reader > the effort of having to download the software and do the same > experiments and analyze the copious output. If they would like to save us effort, they could do so by presenting the proof and eliminating the trust factor. That would certainly make it easier. > They did not present a mathematical > proof of UHP. > They didn't even present the text > of the output of their system when given a suitably symbolized > version of UHP. > The paper in question > convinced [the ATP community], not by proving UHP > or by giving the output/the proof > (way, way too long) No sir. Anything can be abstracted to a shorter higher level. (BTW the proof would not be the output, but rather the logical argument that it is mapped into.) > Their paper does not give a > -mathematical- proof, either in use or mention. On looking at the paper > very closely, I agree with you in principle. You're right, there's no > proof of -any- kind in that paper. > But that paper (and what else I > know about the Boyer-Moore prover) certainly convinces me that they have > done it, been able to prove it automatically (with input describing the > HP situation). Ok. So what proof would be created by their system? > I vaguely recall that you claim to have written a theorem proving system > that proves UHP. Excellent!! As cool as that is, > it is old news. Wouldn't actually showing such a proof be new (since they didn't)? > Oh.. to your original point.. > If one presents a formalism but doesn't show that it represents a > particular logical argument, then they have not shown that the > formalism has any mathematical significance. > Yes, I agree, in some very general sense. But the paper in > question is definitely -not- self-contained. One would need > to be acquainted with the rest of their work to be convinced > that their formalism has mathematical significance. So where is that particular logical argument contained in their other papers? I agree with all of the above except the final conclusion that the proof does exist somewhere. I do thank you for reading the paper and verifying my contention that Boyer and Moore do not give the proof alluded to in the title of their paper. :) C-B > -- > Mitch Harris > (remove q to reply) === Subject: Re: Crossword clue <5H8be.4752$8d4.235@fe1.news.blueyonder.co.uk> Interesting experiment: Come to Texas and see how many > people you can call yank before we have to fire up Ol' Sparky. Interesting learning experience: Realise that some words have different meanings in English and American (qv 'table', 'momentarily', 'pissed', etc etc) (This being Usenet, I fully expect you to say you knew that, and were joking) -- Larry Lard Replies to group please === Subject: Re: Crossword clue <5H8be.4752$8d4.235@fe1.news.blueyonder.co.uk> , Bart Goddard >> huh? get your facts straight... >> troll. >> *plonk* >Like we needed more evidence that anglos have no >sense of humor. > What's an anglo -is it half an Anglo-Saxon? Anglo is an ethnic slur. --- Christopher Heckman === Subject: Re: Crossword clue <5H8be.4752$8d4.235@fe1.news.blueyonder.co.uk> In message , Bart Goddard > huh? get your facts straight... troll. > *plonk* >Like we needed more evidence that anglos have no >>sense of humor. >> What's an anglo -is it half an Anglo-Saxon? >Anglo is an ethnic slur. > --- Christopher Heckman Seems a pretty poor attempt at racism - I thought the good citizens of the USA were meant to be pretty good in this area? -- Jeremy Boden === Subject: Correct proof involving relation theorem? know if the following proof is correct and formal enough for most mathematicians. I'm hoping the logic is sound, though it seems to be. The notation is that [x] is a relation class with the relation being known from the context. That is R[x] = {y in B | xRy} where R subset AxB. Theorem 5.3.3. Let A be a non-empty set, and let ~ be an equivalence relation on A. Let x, y in A. (i) If x ~ y, then [x] = [y]. (ii) If x not~ y, then [x] intersect [y] = nullset. Proof (i). Let a in [x], then x ~ a and so a ~ x. If x ~ y, then a ~ y and so y ~ a. Thus a in [y]. Hence [x] subset [y]. Let a in [y], then y ~ a. If x ~ y, then x ~ a. Thus a in [x]. Hence [y] subset [x]. Since [x] subset [y] and [y] subset [x], we have [x] = [y]. Proof (ii). We prove this by contradiction. Assume [x] intersect [y] not equal nullset. Let a in ([x] intersect [y]), then a in [x] and a in [y]. So x ~ a, and y ~ a, which implies x ~ y, a contradiction. Therefore [x] intersect [y] = nullset. qed. The proof of (ii) might have a logic error, although it seems fine since I assume that the set of [x] intersect [y] has elements and so can assume a is one of them. It's a proof by contradiction. === Subject: Re: Correct proof involving relation theorem? >know if the following proof is correct and formal enough for most >mathematicians. I'm hoping the logic is sound, though it seems to be. >The notation is that [x] is a relation class with the relation being >known from the context. That is R[x] = {y in B | xRy} where R subset AxB. >Theorem 5.3.3. Let A be a non-empty set, and let ~ be an equivalence >relation on A. Let x, y in A. >(i) If x ~ y, then [x] = [y]. >(ii) If x not~ y, then [x] intersect [y] = nullset. >Proof (i). > Let a in [x], then x ~ a and so a ~ x. If x ~ y, then a ~ y and so y ~ >a. Thus a in [y]. Hence [x] subset [y]. > Let a in [y], then y ~ a. If x ~ y, then x ~ a. Thus a in [x]. Hence >[y] subset [x]. That's fine, but instead of repeating the argument it would be better to either say Similarly, [y] subset [x] (or better yet point out that since you've shown that x ~ y implies [x] subset [y] it _follows_ that x ~ y implies [y] subset [x], because...) I say that's better because it's easier to read, and also it shows that you _noticed_ that it's the same argument. > Since [x] subset [y] and [y] subset [x], we have [x] = [y]. >Proof (ii). > We prove this by contradiction. Assume [x] intersect [y] not equal >nullset. Let a in ([x] intersect [y]), then a in [x] and a in [y]. So x >~ a, and y ~ a, which implies x ~ y, a contradiction. Therefore [x] >intersect [y] = nullset. qed. >The proof of (ii) might have a logic error, although it seems fine since >I assume that the set of [x] intersect [y] has elements and so can >assume a is one of them. It's a proof by contradiction. The proof of (ii) seems perfect. ************************ David C. Ullrich === Subject: Re: Correct proof involving relation theorem? > That's fine, but instead of repeating the argument it would be better > to either say Similarly, [y] subset [x] (or better yet point out > that since you've shown that x ~ y implies [x] subset [y] it > _follows_ that x ~ y implies [y] subset [x], because...) I noticed it was the same process, but wanted to show it. I'm not used to writing similarly in my proofs at this point. > The proof of (ii) seems perfect. Excellent. === Subject: a matrix problem with rank Assume that the matrices are all real matrices. A is an m x n matrix, B is an n x s matrix. rank A = rank AB. Show that there exists an s x n matrix C, such that A = ABC. === Subject: matrix problem with rank You may assume that the matrices are all real matrices. A is an m x n matrix, B is an n x s one. rank A = rank AB. Show that there exists an s x n matrix C, such that A = ABC. === Subject: matrix problem with rank You may assume that the matrices are all real matrices. A is an m x n matrix, B is an n x s one. rank A = rank AB. Show that there exists an s x n matrix C, such that A = ABC. === Subject: matrix problem with rank You may assume that the matrices are all real matrices. A is an m x n matrix, B is an n x s one. rank A = rank AB. Show that there exists an s x n matrix C, such that A = ABC. === Subject: Re: matrix problem with rank >You may assume that the matrices are all real matrices. >A is an m x n matrix, B is an n x s one. rank A = rank >AB. >Show that there exists an s x n matrix C, such that A = >ABC. from the assumption that rank A = rank (A*B), it follows that Image(A) = Image (A*B). (Image (A*B) subset Im(A) is clear, and dim (Image(A*B)) = dim (Image(A)) by assumption). Let e_1,...,e_n be the standard basis in IR^n. Since A*e_i is a vector in Image(A) for i=1,...,n and Image (A) = Image (A*B), there exist c_1,...,c_n in IR^s: A*e_i = A*B*c_i. Thus the matrix with the c_i as column vectors has the desired property. Best wishes Torsten. === Subject: Re: matrix problem with rank > You may assume that the matrices are all real matrices. > A is an m x n matrix, B is an n x s one. rank A = rank AB. Clearly you have left out some conditions. Consider: A = (1 0) (0 1) rank(A) = 2 B = (0 0) (0 0) rank(B) = 0 AB = (0 0) (0 0) rank(AB) = 0 != rank(A). - Randy === Subject: Re: matrix problem with rank rank A = rank AB is a given assumption... === Subject: Re: matrix problem with rank > rank A = rank AB > is a given assumption... Ah, OK. I thought that was a theorem you were supposed to prove. Perhaps this can get you started: If B is square, then rank A = rank AB if and only if B is non-singular. But of course if B is non-singular, then you can choose C = inv(B), giving A = ABC. Now you are asked to examine the non-square case. Look for the proof of the square result, that may give you some hints for what parts generalize to the non-square case in this way (note that A = ABC doesn't mean that C is a right inverse of B, BC = I. It's weaker than that.) - Randy === Subject: Re: matrix problem with rank > You may assume that the matrices are all real matrices. > A is an m x n matrix, B is an n x s one. rank A = rank AB. > Show that there exists an s x n matrix C, such that A = ABC. Sounds very much like a homework problem or - since you posted it several times - a punishment. In both cases you are assumed to do it yourself. Alois === Subject: Re: matrix problem with rank I am sorry about the duplicated posting. That is the network problem here. It always said page cannot display, so I posted it again. === Subject: Re: What is the greatest error ever committed in math? > i wanna know...please help... A tragedy if not error, for non-Euclidean geometry... When cowed down by Gauss, Bolyai Junior should not have left maths, especially when Bolyai Senior could have averted it. Or Ramanujan returning to India only to die,when so much nourishing food was available in England, could slightly compromise religious views ... === Subject: Re: What is the greatest error ever committed in math? >One long-standing error was the belief that a loxodrome >curve( http://mathworld.wolfram.com/Loxodrome.html ) was >the shortest distance between two points on the Earth's >surface. Only on a Mercator projection, lines of constant compass setting cut the rectangular grid lines representing latitude/longitude at a constant angle.. as shortest distance lines. On the globe, best aproximation to geodesy is at the equator and the worst at poles. === Subject: Re: What is the greatest error ever committed in math? > i wanna know...please help... Arguably another great error was the widespread belief or assumption in the 18th century that every continuous function was diferentiable. But I'm not a historian, and it could be this statement is itself an error or oversimplification. === Subject: Re: What is the greatest error ever committed in math? >Surely the greatest mistake in maths is committed every time someone >or 5^100. Having in front of me a large pile of exams filled with that kind of error, I can tell you it's not so great... Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: 1-1 curve with lebesque measure greater than 0 >>can anyone think of a 1-1 curve with lebesque measure greater than 0? >>is is probably some sort of modification of a space filling curve >> Well such things certainly exist. >> For example you could start with a curve with a lot of vertical >> zig-zags. Then at the next stage each vertical segment is replaced >> by a curve with much shorter and much more densely packed horizonal >> zigzags, at the next stage each of those horizontal segments is >> replaced by more vertical zigzags, etc. If you do that right the >> limit should be a 1-1 curve with positive area. >> ************************ >> David C. Ullrich >I wonder if it does not clash with the fact that the planar Lebesgue >measure of the graph of a Lebesgue measurable function is zero. The approximating curves are not graphs of functions, not that that matters since yes, they do have zero area. >At each >step above, if you keep the curve 1-1, you keep the measure at 0, since >Lebesgue measure is sigma-additive. Or did I miss something? You missed something. Not that it's really fair to put it that way, since all I gave was a vague sketch, but your objections are no problem. Yes, at each stage of the construction we have a curve with area 0, but the curve we finally construct is the _limit_ of those curves, not the union. I'm not going to try to describe things more precisely and prove that the limit has positive area in a usenet post. Instead I'll just point out that the standard constructions of space-filling curves construct them as limits of curves with area 0 - if your objection were valid it would apply to those space-filling curves as well, ultimately leading to a proof that the plane has Lebesgue measure 0. ************************ David C. Ullrich === Subject: Re: 1-1 curve with lebesque measure greater than 0 >can anyone think of a 1-1 curve with lebesque > measure greater than 0? >is is probably some sort of modification of a space > filling curve > Yes, there are such. One way to construct one is to > consider > two 'fat' Cantor sets in [0,1], i.e. Cantor sets of > strictly > positive measure. In [0,1]^2, define your curve as a > limit of > curves that go through all the 'corners' obtained > from endpoints > of each stage of the construction of the Cantor sets. > With care, > you obtain a 1-1 curve going through all the points > in the cross > product of the Cantor sets, so the curve has measure > more than 0. > --Dan Grubb === Subject: Re: Question about the minesweeper consistency problem (NP-complete problems) > What are you really after? Well, believe it or not, but I've developed an algorithm which solves the minesweeper consistency problem and I think it runs in polynomial time. I haven't made a formal complexity analysis like you see in complexity theory texts (in which you consider every instruction etc), course I did make some calculations in order to calculate the maximum number of iterations of the core block of the program (whose running time, in a single iteration, grows polynomially, and also weakly, in fact if it wasn't for the logarithmically growing size of cell identifiers there would be a worst case running time Tr which wouldn't be dependent on the size of the input string). If we consider Tr as the constant worst case running time of the core algorithm block, the algorithm complexity would be theta(n^3). I tested the program, it works and gives exact results, and it takes 15 seconds in order to process a game situation corresponding to the expert level of the real game. It included something like 134 uncovered cells adjacent to covered cells (which are the only sources of information, covered cells or uncovered cells which are adjacent to no covered cells don't add information, they don't add conditions to be respected by the mines distribution which could be a possible solution). This means that my program is certainly sub-exponential, but I can also tell you that it NEVER considers a single entire possible solution while it runs. When the algorithm ends you don't have a possible solution, but you can build a random possible solution using extra (polynomial) time (if the solution is unique, of course you can build only that solution). By the way, do you have a good link describing those optimized solvers you talked about? However, if I understand correctly what you say, I think that those solvers test entire possible solutions (they simply avoid to test ALL the possible solutions), so in that case my algorithm is certainly different. === Subject: Re: Question about the minesweeper consistency problem (NP-complete problems) Well, believe it or not, but I've developed an algorithm which solves > the minesweeper consistency problem and I think it runs in polynomial > time. I haven't made a formal complexity analysis like you see in Well, I don't believe it then --- because, as you well know, the minesweeper problem is NP-complete. > I tested the program, it works and gives exact results, and it takes > 15 seconds in order to process a game situation corresponding to the > expert level of the real game. It included something like 134 > uncovered cells adjacent to covered cells (which are the only sources > of information, covered cells or uncovered cells which are adjacent to > no covered cells don't add information, they don't add conditions to > be respected by the mines distribution which could be a possible > solution). > This means that my program is certainly sub-exponential, but I can Why does it? I've played the minesweeper game, and my impression is that those configurations which account for the NP-completeness are actually not that common. So if you tested your program on random configurations, it may have been that you just weren't likely to hit hard situations. Also, it may be that you need to increase the playing field further to really encounter these difficulties. (Perhaps a transition point analysis can be used to generate 'hard' instances ...) So likely your algorithm doesn't run in polynomial time (but ran quickly on the instances you considered), or it isn't always right (but was right on the instances you tested). (Or both. :P) Lasse --- === Subject: Back Online We are back online: www.mathematician.org I'm on vacation for the next 3 weeks, so development of the website will be nil. Sorry. Webmaster - Mathematician.org === Subject: What's the name for this technique? The other day in class I was showing how one can do analytic continuation of the zeta function beyond the region where the traditional series converges. At one point we have the integral [i] int_0^1 1/(e^t-1) t^{z-1} dt, which converges only for Re(z) > 1, and we rewrite this as int_0^1 (1/(e^t-1) - 1/t) t^{z-1} dt + int_0^1 1/t t^{z-1} dt [ii] = int_0^1 (1/(e^t-1) - 1/t) t^{z-1} dt + 1/(z-1), where the integral in the last expression converges for Re(z) > 0. I pointed out that exactly this technique can be useful in numerical analysis, for example if we wanted to evaluate zeta(3/2) numerically then using [ii] would be better than [i] because the integrand in [ii] is bounded (note below). My question is this: What's the _name_ for this technique in numerical analysis? I could swear I've seen this mentioned in some book, as the ___ method. (Note: yes, of course evaluating the integrand in [ii] accurately could be a problem because we have the difference of two huge numbers - probably one would want to massage it a little: 1/(e^t-1) - 1/t = ((1+t) - e^t)/(t(e^t-1)). That's better, since it only involves the difference of medium-sized numbers; maybe replacing e^t-1 and 1+t-e^t by power series would be better yet?) ************************ David C. Ullrich === Subject: Re: What's the name for this technique? > What's the _name_ for this technique in numerical > analysis? I could swear I've seen this mentioned > in some book, as the ___ method. Francis Scheid simply calls it 'subtracting the singularity' in the first edition of his Schaum's Outline. -- write(*,*) transfer((/17.392111325966148d0,6.5794487871554595D-85, & 6.0134700243160014d-154/),(/'x'/)); end === Subject: Re: What's the name for this technique? > The other day in class I was showing how one can do > analytic continuation of the zeta function beyond > the region where the traditional series converges. > At one point we have the integral > [i] int_0^1 1/(e^t-1) t^{z-1} dt, > which converges only for Re(z) > 1, and we rewrite > this as > int_0^1 (1/(e^t-1) - 1/t) t^{z-1} dt > + int_0^1 1/t t^{z-1} dt > [ii] = int_0^1 (1/(e^t-1) - 1/t) t^{z-1} dt + 1/(z-1), > where the integral in the last expression converges > for Re(z) > 0. > I pointed out that exactly this technique can be > useful in numerical analysis, for example if we > wanted to evaluate zeta(3/2) numerically then using > [ii] would be better than [i] because the integrand > in [ii] is bounded (note below). My question is this: > What's the _name_ for this technique in numerical > analysis? I could swear I've seen this mentioned > in some book, as the ___ method. I am not quite sure if you are referring to Romberg's method of numerical integration mentioned in a textbook* from my college years. The book uses Bernoulli numbers to evaluate zeta function particularly for calculating inherent error in integration by Romberg's method. * Numerical mathematics & computing - second edittion, Cheny and Kincaid, pp176 -- Respectfully, Mohan Pawar MIO Instruments LLC (920) 277-6037 > (Note: yes, of course evaluating the integrand in > [ii] accurately could be a problem because we have > the difference of two huge numbers - probably one > would want to massage it a little: > 1/(e^t-1) - 1/t = ((1+t) - e^t)/(t(e^t-1)). > That's better, since it only involves the > difference of medium-sized numbers; maybe > replacing e^t-1 and 1+t-e^t by power series > would be better yet?) > ************************ > David C. Ullrich === Subject: Re: What's the name for this technique? >The other day in class I was showing how one can do >analytic continuation of the zeta function beyond >the region where the traditional series converges. >At one point we have the integral >[i] int_0^1 1/(e^t-1) t^{z-1} dt, > >which converges only for Re(z) > 1, and we rewrite >this as > int_0^1 (1/(e^t-1) - 1/t) t^{z-1} dt > > + int_0^1 1/t t^{z-1} dt > >[ii] = int_0^1 (1/(e^t-1) - 1/t) t^{z-1} dt + 1/(z-1), > >where the integral in the last expression converges >for Re(z) > 0. >I pointed out that exactly this technique can be >useful in numerical analysis, for example if we >wanted to evaluate zeta(3/2) numerically then using >[ii] would be better than [i] because the integrand >in [ii] is bounded (note below). My question is this: >What's the _name_ for this technique in numerical >analysis? I could swear I've seen this mentioned >in some book, as the ___ method. >(Note: yes, of course evaluating the integrand in >[ii] accurately could be a problem because we have >the difference of two huge numbers - probably one >would want to massage it a little: > 1/(e^t-1) - 1/t = ((1+t) - e^t)/(t(e^t-1)). > >That's better, since it only involves the >difference of medium-sized numbers; maybe >replacing e^t-1 and 1+t-e^t by power series >would be better yet?) yes clearly near t=0 you will need to use the power series of ((1+t) - e^t)/(t(e^t-1)) in order to avoid serious effects from >************************ >David C. Ullrich i teach this as the technique of splitting off the singular part in an improper integral, a technique used often in BEM (which works only if this part is analytically integrable.) hth peter === Subject: Re: What's the name for this technique? >The other day in class I was showing how one can do >analytic continuation of the zeta function beyond >the region where the traditional series converges. >At one point we have the integral >[i] int_0^1 1/(e^t-1) t^{z-1} dt, > >which converges only for Re(z) > 1, and we rewrite >this as > int_0^1 (1/(e^t-1) - 1/t) t^{z-1} dt > > + int_0^1 1/t t^{z-1} dt > >[ii] = int_0^1 (1/(e^t-1) - 1/t) t^{z-1} dt + 1/(z-1), > >where the integral in the last expression converges >for Re(z) > 0. >I pointed out that exactly this technique can be >useful in numerical analysis, for example if we >wanted to evaluate zeta(3/2) numerically then using >[ii] would be better than [i] because the integrand >in [ii] is bounded (note below). My question is this: >What's the _name_ for this technique in numerical >analysis? I could swear I've seen this mentioned >in some book, as the ___ method. > [...] >i teach this as the technique of splitting off the singular part >in an improper integral, a technique used often in BEM >(which works only if this part is analytically integrable.) >hth >peter ************************ David C. Ullrich === Subject: Re: What's the name for this technique? analytic continuation of the zeta function beyond >the region where the traditional series converges. >At one point we have the integral [i] int_0^1 1/(e^t-1) t^{z-1} dt, which converges only for Re(z) > 1, and we rewrite >this as int_0^1 (1/(e^t-1) - 1/t) t^{z-1} dt + int_0^1 1/t t^{z-1} dt [ii] = int_0^1 (1/(e^t-1) - 1/t) t^{z-1} dt + 1/(z-1), where the integral in the last expression converges >for Re(z) > 0. I pointed out that exactly this technique can be >useful in numerical analysis, for example if we >wanted to evaluate zeta(3/2) numerically then using >[ii] would be better than [i] because the integrand >in [ii] is bounded (note below). My question is this: What's the _name_ for this technique in numerical >analysis? I could swear I've seen this mentioned >in some book, as the ___ method. > [...] >i teach this as the technique of splitting off the singular part >in an improper integral, a technique used often in BEM >(which works only if this part is analytically integrable.) >hth >peter BEM = boundary element method, a variation on the finite element method in which (due to existence of a Green's function for some constant coefficient PDE) we can discretize the boundary of the domain rather than the interior. These methods invariably(?) involve a (benign) singular integral over the boundary. Peter's terminology is nice; I was going to nominate removing the singularity but this probably echoes removable singularity a bit too strongly. === Subject: Re: prove surjection >R modulo I... Are you now being ironic ? I wonder why the fact that R/I stands for (better: can be called) R modulo I should be helping you. What is needed is to remember that this is a quotient of the set R by an equivalence relation that uses I (which one ?) and that such a relation defines equiv. classes in R, and all that ! === Subject: Re: Some Simple Questions >>Which is it - can you give the proof that they produced, or do you >>agree that you don't see one either? >> Neither. >I think not. You have stated that you are not relying on any such >proof having been produced, but rather on the claims of others. So the >latter alternative is the case (you don't see one either.) No, neither. I can't give the proof they produced (I'm not particularly interested in reproducing their results. Particularly for something that is already well known.) But, I can see how they produced it. >The question is whether Boyer and Moore have produced such a proof, and >we agree that neither of us sees such a proof. That is my point. >If you want to believe that such a proof exists somewhere, that is your >prerogative. But from an academic point of view, the question is >whether such a proof has been given. I've read the paper now, and Mitch Harris (in another thread) gave you a better explanation then I ever could. Martin === Subject: Re: Some Simple Questions <8ovu61t73getr6ekspeh8foannr0c9vodd@4ax.com> <4qg171p68dncodciimtjtohtoatqlm21a0@4ax.com>Which is it - can you give the proof that they produced, or do you >>agree that you don't see one either? >> Neither. >I think not. You have stated that you are not relying on any such >proof having been produced, but rather on the claims of others. So the >latter alternative is the case (you don't see one either.) > No, neither. I can't give the proof they produced It sounds like you don't see it. Do you know where it is located? > (I'm not > particularly interested in reproducing their results. Particularly > for something that is already well known.) Where that elusive proof is is not very well-known, IMHO. Nobody has been able to point it out. C-B > Martin === Subject: JSH: SFT is boring You may find James' pathology exotic. Someone like myself, whose day job is teaching in an inner-city high school, sees it all the time. === Subject: Re: JSH: SFT is boring > You may find James' pathology exotic. Someone like myself, whose day job is > teaching in an inner-city high school, sees it all the time. Interesting. Could you please elaborate a bit on that? Jose Carlos Santos === Subject: Re: JSH: SFT is boring >> You may find James' pathology exotic. >> Someone like myself, whose day job >> is teaching in an inner-city high school, >> sees it all the time. > Interesting. Could you please elaborate a bit on that? A few desperate teenagers, trapped in a poisoned world, act out. They are in denial; narcissistic; no humility; sorry for themselves, claiming to be misunderstood; a lack of patience, accompanied by an unrealistic sense of urgency; are manipulative: anything to get their way; are unstable: sometimes have an exaggerated sense of self importance, at other times are terribly insecure; make doomsday prophesies, sound the alarm like Chicken Little; strike out at others, whom they perceive to be getting a free ride or having it better; do not respect boundaries but are quick to react, if they perceive their territory to be violated. Yuck. === Subject: Re: JSH: SFT is boring >You may find James' pathology exotic. >Someone like myself, whose day job >is teaching in an inner-city high school, >sees it all the time. >>Interesting. Could you please elaborate a bit on that? > A few desperate teenagers, trapped in a poisoned world, > act out. They are in denial; narcissistic; no humility; sorry > for themselves, claiming to be misunderstood; a lack of > patience, accompanied by an unrealistic sense of urgency; > are manipulative: anything to get their way; are unstable: > sometimes have an exaggerated sense of self importance, at > other times are terribly insecure; make doomsday prophesies, > sound the alarm like Chicken Little; strike out at others, > whom they perceive to be getting a free ride or having it > better; do not respect boundaries but are quick to react, > if they perceive their territory to be violated. Yuck. Jose Carlos Santos === Subject: Re: good text for linear algebra? > Can someone suggest a good text book for linear algebra? One suggestion > made to me was Linear algebra done right by Sheldon Axler. Ouch! I guess it depends on why your students are learning linear algebra... My only experience with Linear Algebra Done Right: I was teaching our honors differential equations course. The previous term they studied linear algebra. So we discussed linear differential equations, linear independence of solutions, Wronskians. At one point I said: This system of linear equations can be solved by Cramer's Rule. After class one of the students informed me that none of the students in the class had ever heard of Cramer's Rule. I found that hard to believe. So I find out what text they used the previous quarter. Yes, indeed, it was Linear Algebra Done Right. I got a copy of the book to check. No mention of Cramer's Rule. In fact, not even any mention of determinants until the last 20 pages of the book. So in the end I gave the differential equations class some exercises on determinants and Cramer's Rule. Now, whether linear algebra had been done right I cannot say, but at least it was not done well enough for applying linear algebra to subsequent courses... -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: good text for linear algebra? <280420050759278066%edgar@math.ohio-state.edu.invalid Can someone suggest a good text book for linear algebra? One suggestion > made to me was Linear algebra done right by Sheldon Axler. > Ouch! I guess it depends on why your students are learning linear > algebra... > My only experience with Linear Algebra Done Right: > I was teaching our honors differential equations course. The previous > term they studied linear algebra. So we discussed linear differential > equations, linear independence of solutions, Wronskians. At one point > I said: This system of linear equations can be solved by Cramer's > Rule. After class one of the students informed me that none of the > students in the class had ever heard of Cramer's Rule. I found that > hard to believe. So I find out what text they used the previous > quarter. Yes, indeed, it was Linear Algebra Done Right. I got a > copy of the book to check. No mention of Cramer's Rule. In fact, not > even any mention of determinants until the last 20 pages of the book. > So in the end I gave the differential equations class some exercises on > determinants and Cramer's Rule. Now, whether linear algebra had been > done right I cannot say, but at least it was not done well enough for > applying linear algebra to subsequent courses... And I thought Cramer's rule (and determinants!) would still be covered in a decent second-year high school algebra course. Sigh. === Subject: Re: good text for linear algebra? > So in the end I gave the differential equations class some exercises > on > determinants and Cramer's Rule. Now, whether linear algebra had been > done right I cannot say, but at least it was not done well enough > for > applying linear algebra to subsequent courses... > And I thought Cramer's rule (and determinants!) would still be covered > in a decent second-year high school algebra course. Sigh. i ran into determinants in three different classes (algebra II, calc III, calc V) that were prerequisities for ODE class. michael === Subject: Re: good text for linear algebra? >My only experience with Linear Algebra Done Right: ... >In fact, not >even any mention of determinants until the last 20 pages of the book. >So in the end I gave the differential equations class some exercises on >determinants and Cramer's Rule. Now, whether linear algebra had been >done right I cannot say, but at least it was not done well enough for >applying linear algebra to subsequent courses... Well, of course. The proper prerequisite(s) for *those* are its sequels, Bilinear Algebra Done Right (for applications to quadratic forms, etc.) and Multilinear Algebra Done Right (where you finally get to determinants!). Lee Rudolph === Subject: Re: good text for linear algebra? Definitely Sheldon Axler's book. His approach is to teach linear algebra via linear transformations on vector spaces; it's not until the later chapters that he introduces matrices (as representations of linear transformations after a choice of basis). Also, the chapter on eigenvalues comes before his chapter on determinants! You won't learn about Gauss-Jordan or any of the pure matrix manipulation stuff (like how to create a basis from a linearly independent set of vectors, how to calculate inverses, etc), you _will_ learn about linear algebra, however :) Willem > Can someone suggest a good text book for linear algebra? One suggestion > made to me was Linear algebra done right by Sheldon Axler. Any > comments about it? Though I have never taken a course on linear algebra, > I am quite familiar with the basics like working with matrices etc. === Subject: Re: good text for linear algebra? >Definitely Sheldon Axler's book. His approach is to teach linear algebra >via linear transformations on vector spaces; it's not until the later >chapters >that he introduces matrices (as representations of linear transformations >after a choice of basis). Also, the chapter on eigenvalues comes before >his chapter on determinants! The fact that you think an exclamation mark belongs here is symptomatic of the prevailing mentality that Axler wants to overcome. Eigenvalues (and singular values for that matter) are much more central to the ideas of linear algebra than the existence of a multiplicative map M_n(R) --> R . My experience with determinants and eigenvalues is akin to my experience with integrals and antiderivatives. We all go through Riemann sums and whatnot to explain what integrals _are_. Five minutes after having shown students the Fundamental Theorem of Calculus, none of them can distinguish integrals from antiderivatives any more, and they will tell you that e.g. int_R exp(-x^2) dx can't be done. So too with eigenvalues. Sample question: ask them to compute the eigenvalues of a diagonal matrix (after they've seen that eigenvalues _may_ be computed as roots of det(A-XI) ). Do you really think they understand what eigenvalues _are_ any more? dave === Subject: Re: Axiomatic derivation of complex number system > I am wanting to understand a complete derivation of the complex > number exponentiation, division functions at a detailed level. > Once you've defined addition and multiplication in the usual way, > exponentiation is trivial, namely exp(z) is given by the same power > series you learned for real values of z, and division is simply the > solution to a simple multiplication equation. What more do you need?? > I wanted to know how exponentiation is derived for C, not just exp(z). > I suspect exponentiation is derived through exp(z). Definition: z^w = exp(w*log(z)) === Subject: Re: Groups of order 32 I also just found this on the web, though I don't like a lot of the computer printouts of stuff about groups of order 32--I don't know anything about the things like the cohomology groups etc. in the magma and GAP results that can be found. At the URL http://mail.math.ucsb.edu/~brian/groups.html # Order 32 (7 abelian, 45 nonabelian): 1. Z32 2. Z2 + Z16 3. Z4 + Z8 4. Z2 + Z2 + Z8 5. Z2 + Z4 + Z4 6. Z2 + Z2 + Z2 + Z4 7. Z2 + Z2 + Z2 + Z2 + Z2 8. D16 9. Q8 10. Q2 x Z4 11. Q2 x Z2 x Z2 12. D4 x Z4 13. D4 x Z2 x Z2 14. Q4 x Z2 15. D8 x Z2 16. (gp of order 16 #10) x Z2 17. (gp of order 16 #11) x Z2 18. (gp of order 16 #12) x Z2 19. (gp of order 16 #13) x Z2 20. (gp of order 16 #14) x Z2 21. 22. 23. 24. 25. 26. 27. (and a whole lot more!!) ---------- There are only 51 (not 52) though. Van === Subject: Re: Groups of order 32 > I just barely started on this, and have not yet read anything about > it (I will look at past posts here). > First, the groups with an element x of order |x| = 16, so N = is > normal, and G = ; x^16 = 1 = y^2. > yxy^-1 = x^r ; r in Z*_16 = Z_2 x Z_4, so there are 8 values of r. If x^16 = y^2 = 1 and yxy^-1 = x^r, then we must have r^2 = 1 (mod 16). > For elements of order 8 in groups of order 16, all 4 values of r gave > different > groups, and I think that this may give 8 different groups here, though > I haven't yet shown this. ?? You've lost me. > I think showing that the permutations of the exponents > x^k being different means that the groups are different, but if this is > true, I don't know how to show it. If one group has an element x of order n and an element y of order m with yxy^-1 = x^r, and another group doesn't, then they can't be isomorphic. > r = 1,3,5,...13,15 = +/- 1,3,5,7 BTW, could please fix your line length? I've done so for some, but not all, of your post above. I suspect you've set your newsreader to wrap at a certain length upon sending, but are typing your lines with a longer length. -- Jim Heckman === Subject: Re: Groups of order 32 <1171hefbjlb8qba@corp.supernews.com I just barely started on this, and have not yet read anything about > it (I will look at past posts here). > First, the groups with an element x of order |x| = 16, so N = is > normal, and G = ; x^16 = 1 = y^2. > yxy^-1 = x^r ; r in Z*_16 = Z_2 x Z_4, so there are 8 values of r. > If x^16 = y^2 = 1 and yxy^-1 = x^r, then we must have > r^2 = 1 (mod 16). This is basic, of course. I am ashamed to have missed this. This means that the 4 values +/- 3,5 = (3,5,11,13) mod 16 don't work, so we have the 4 values 1,7,9,15 mod 16 for r. r = 9 has yx^2y = x^2 so the center is Z = =~ Z_8; G/Z = Z_2 x Z_2. r = 7 has Z = like D_16 with r = - 1, but 7Z_16 = (0,7,14,5,12,3,10,1,8,15,6,13,4,11,2,9) is very different from 15Z_16 = -Z_16, so these 4 groups are all different. As I said, there is also the generalized quaternion group of order 32, and I don't think there are any other groups (except Z_32) with elements of order 16. >>For elements of order 8 in groups of order 16, all 4 values of r gave >>different >> groups, and I think that this may give 8 different groups here, though >>I haven't yet shown this. > ?? You've lost me. Yes, this makes no sense. >>I think showing that the permutations of the exponents >>x^k being different means that the groups are different, but if this is >>true, I don't know how to show it. > If one group has an element x of order n and an element y of > order m with yxy^-1 = x^r, and another group doesn't, then they > can't be isomorphic. > r = 1,3,5,...13,15 = +/- 1,3,5,7 > BTW, could please fix your line length? I've done so for some, > but not all, of your post above. I suspect you've set your > newsreader to wrap at a certain length upon sending, but are > typing your lines with a longer length. > -- > Jim Heckman I have noticed the problem with line length too. I will shorten my should, as having a couple characters left on the next line makes a mess of my posts. This post is cleaned up somewhat, although the quotes make some problems. Van === Subject: Re: Groups of order 32 <1171hefbjlb8qba@corp.supernews.com> I found this on the web, though I don't like a lot of the computer printouts of stuff about groups of order 32, mainly because I don't know anything about the things like the cohomology groups etc. in the magma and GAP results that can be found at the URL http://mail.math.ucsb.edu/~brian/groups.html I have added some notes myself. ----------- Order 32 (7 abelian, 45 nonabelian) (Should be 44 nonabelian): 1. Z32 2. Z2 + Z16 3. Z4 + Z8 4. Z2 + Z2 + Z8 5. Z2 + Z4 + Z4 6. Z2 + Z2 + Z2 + Z4 7. Z2 + Z2 + Z2 + Z2 + Z2 These are the 7 Abelian groups. 8. D16 (yxy = x^-1 = x^15). 9. Q8 = (I think that's what is meant here.) 10. Q2 x Z4 (Q2 must be the quaternion group of order 8). 11. Q2 x Z2 x Z2 12. D4 x Z4 13. D4 x Z2 x Z2 14. Q4 x Z2 (Q4 must be the 16 element generalized quaternions; Q4 = 15. D8 x Z2 16. (gp of order 16 #10) x Z2 17. (gp of order 16 #11) x Z2 18. (gp of order 16 #12) x Z2 19. (gp of order 16 #13) x Z2 20. (gp of order 16 #14) x Z2 I don't know where this numbering of groups comes from. Maybe GAP or MAGMA. 21. 22. 23. 24. 25. 26. 27. (and a whole lot more!!) In particular, 3 more with yxy = x^r; r = 1,7,9,15 = +/- 1,7, as then r^2 = 1. 3 and 5 have order 4, so they don't work. This post previews OK--I hope line length works. Van === Subject: Re: Groups of order 32 I forgot the generalized quaterions of order 32 given by Q_5 = I think for n = 2^m, Q_m = and yxy^-1 = x^-1 Van === Subject: Good reduction of Jacobians of twisted Fermat curves question about twisted Fermat curves. Let p > 3 be a prime number, and let a,b,c be pairwise coprime integers not divisible by p. Let l neq p be a prime dividing abc. Let F be the curve a x^p + bx^p + cx^p. and let J be its Jacobian. Are there inegers a,b,c such that for all l | abc the Jacobian J has good reduction at l ? T. === Subject: Re: Good reduction of Jacobians of twisted Fermat curves that's all one, divisor of a times b times c?... what is 1 neq p? >b) the hare-brained analysis of WTC7 assumes that the substructure of the complex, which included a large subway station, was not affected by the two largest buildings, falling down into it. --Chair Man George NAND Strep Throat @ W'gate? http://tarpley.net/bush12.htm http://laroucehpub.com http://members.tripod.com/~ame[CapitalEth]rican almanac === Subject: Is there a Platonistic interpretation of the empty set? Many objects can be perceived or considered at once, and this is the foundation for the concept of set. Other than a mark on paper, what, then, is the empty set? === Subject: Re: Is there a Platonistic interpretation of the empty set? Ì Mike H ó.8d.98.87.8b.8c .97.99.95 .92.86.94.9d.92.87 > Many objects can be perceived or considered at once, and this is the > foundation for the concept of set. Other than a mark on paper, what, then, > is the empty set? The question is very general; roughly speaking, in a Platonist (Mathematical) world the empty set is the basis of everything that exists, since using the classic construction: {}=0, {0}=1, {0,1}=2, {0,1,2}=3, ..., one constructs the Natural numbers from it, and that's all that's really needed to construct everything (Mathematical) that exists (in this universe). It is like a seed that generates the entire universe or alternatively, it defines the boundary between existence and non-existence (in such a universe). -- I. N. Galidakis http://users.forthnet.gr/ath/jgal/ Eventually, _everything_ is understandable === Subject: Re: Is there a Platonistic interpretation of the empty set? > Many objects can be perceived or considered at once, and this is the > foundation for the concept of set. Other than a mark on paper, what, then, > is the empty set? In the standard ZFC the only individual is the empty set, although multiple individuals are allowed. When you remove the empty set, nothing else remains obviously. The fact that the empty set is unique and does not have any extension does make the philosopher think of it as a Platonic idea. In the real world, you don't construct things out of nothing, so it really does not talk of our world. If you want a non-Platonist interpretation think of it as not merely play with marks, but with thought experiments about a strange world with strange rules (physics). -- Eray === Subject: Re: Is there a Platonistic interpretation of the empty set? It's a nest, when the birds left. It's the smiling of the Cheshire-cat. Hero === Subject: Re: Is there a Platonistic interpretation of the empty set? > Many objects can be perceived or considered at once, and this is the > foundation for the concept of set. Other than a mark on paper, what, then, > is the empty set? What's left when you remove the objects. === Subject: Re: Is there a Platonistic interpretation of the empty set? > Many objects can be perceived or considered at once, and this is the > foundation for the concept of set. Other than a mark on paper, what, then, > is the empty set? > What's left when you remove the objects. Or: the complement of the universe. Platonistic enough? === Subject: Re: Is there a Platonistic interpretation of the empty set? >> Many objects can be perceived or considered at once, and this is the >> foundation for the concept of set. Other than a mark on paper, what, then, >> is the empty set? >What's left when you remove the objects. I think he was asking, what *is* left when you remove the objects? In any case, while it has many fantastic properties, pondering and speculating about the empty set is a pastime best left outside the classroom (and sci.math). === Subject: rank & eigenvalues [given A - n*n matrix] Is there any connection between rank(xI - A) and the number of eigenvalues of A (including algebric measure) ? PS I hope that algebric measure is the correct name for the dimention of the 'eigenspace'. === Subject: Re: rank & eigenvalues > [given A - n*n matrix] > Is there any connection between rank(xI - A) and the number of > eigenvalues of A (including algebric measure) ? > PS > I hope that algebric measure is the correct name for the > dimention of the 'eigenspace'. Maybe you mean something like geometric multiplicity? Consider the matrix | a 1 0 | | 0 a 0 | | 0 0 a | What is the algebraic measure of this system, what is the rank of (Ax-I) ? Alois -- Alois Steindl, Tel.: +43 (1) 58801 / 32558 Inst. for Mechanics II, Fax.: +43 (1) 58801 / 32598 Vienna University of Technology, A-1040 Wiedner Hauptstr. 8-10 === Subject: Re: rank & eigenvalues A. I understand it's called multiplicity and not measure. 'geometric' definition. C. By algebric I mean the number of times it's seen in the characteristic polynomial. and to the point. why do you ask me about (Ax - I) ? I think it should be (xI - A) [to satisfy Av = xv]. [let A be your matrix, C(A) the char. polynomial] C(A) = (x-a)^3 (3 being the algebric multiplicity of the eigenvalue a :) Va - the sigenspace of a - span{(0,1,0),(0,0,1)} the geometric multiplicity of a is 2 rank (aI - A) = 1 1 + 2 = 3 [rank(aI - A) + dim(Va) = size of A's columns] - maybe it's always like that? === Subject: Re: rank & eigenvalues > A. I understand it's called multiplicity and not measure. > 'geometric' definition. > C. By algebric I mean the number of times it's seen in the > characteristic polynomial. > and to the point. > why do you ask me about (Ax - I) ? I think it should be (xI - A) [to > satisfy Av = xv]. Of course, yes. (I really wanted to type (A-xI), to obtain the usual order) > [let A be your matrix, C(A) the char. polynomial] > C(A) = (x-a)^3 > (3 being the algebric multiplicity of the eigenvalue a :) > Va - the sigenspace of a - span{(0,1,0),(0,0,1)} Shouldn't that be span{(1,0,0),(0,0,1)}? But the dimension is right. > the geometric multiplicity of a is 2 > rank (aI - A) = 1 > 1 + 2 = 3 [rank(aI - A) + dim(Va) = size of A's columns] - maybe it's > always like that? Yes. Alois === Subject: SDE-Vasicek Modell-Solution I'm studying interest rate models and while doing so I found the following SDE in Vasicek Model: dr(t)=k*(omega-r(t))+sigma*dW(t) the solution for every s the field of vision is not *perfectly* circular, > owing primarily to the shape of the blind spot, It has to do with evolution as well (not sure about that blind spot idea). Most of mans prey, as well as that which what would hunt a man, travels in the horizontal plane, along the ground. For the same reason, cats have more movement acuity in the horizontal direction. > which is fortunately not in the straight-ahead focal point. > the only overlap is akin to the apparent libration > of Moon, due to rolling under it (standing > on Earth), but much smaller. > the main lacuna is the nose, and the apparent size depends > upon the closeness of one's eyes ... and number of them! > --Chair Man George XOR Strep Throat @ W'gate? > http://tarpley.net/bush12.htm > http://larouchepub.com > http://members.tripod.com/~american_almanac === Subject: Re: Finding the largest rectangle which fit inside a circle. the blind spot is where the optic nerves opens into the retina; it's normally hard to see, and it's really apparent with eyes closed. as far as I know, the retina is essentially spherical; thus, the field of views is essentially circular. >b) the hare-brained analysis of WTC7 assumes that the substructure of the complex, which included a large subway station, was not affected by the two largest buildings, falling down into it. --Chair Man George NAND Strep Throat @ W'gate? http://tarpley.net/bush12.htm http://laroucehpub.com http://members.tripod.com/~ame[CapitalEth]rican almanac === Subject: Re: How to compute the determinant of a special type Toeplitz matrix Here is another question. What is the inverse of this matrix? >I would like to compute the determinant of an M by M Toeplitz matrix A such > as > row 1: a r r^2 r^3 ... r^(M-1) > row 2: r^(-1) a r r^2 ... r^(M-2) > row 3: r^(-2) r^(-1) a r ... r^(M-3) > row M-1: r^(-(M-2)) r^(-(M-3)) ... a r > row M: r^(-(M-1)) r^(-(M-2)) ... r^(-1) a > where multplication of the elements of the ith row and jth column and > jth row and ith column is always 1 (a_{ij}a_{ji} = 1) and a_{ii} = a > The determinant is > (a-1)^(M-1)(a+M-1) === Subject: Re: How to compute the determinant of a special type Toeplitz matrix I think that I computed the inverse of this matrix as follows. row 1: a+M-2 -r -r^2 ... -r^(M-1) row 2: -r^(-1) a+M-2 -r ... -r^(M-2) row M: -r^(-(M-1)) -r^(-(M-2)) ... -r^(-1) a+M-2 with all elements multiplied by 1/(a-1)(a+M-1) > Here is another question. > What is the inverse of this matrix? >I would like to compute the determinant of an M by M Toeplitz matrix A > such > as row 1: a r r^2 r^3 ... r^(M-1) > row 2: r^(-1) a r r^2 ... r^(M-2) > row 3: r^(-2) r^(-1) a r ... r^(M-3) row M-1: r^(-(M-2)) r^(-(M-3)) ... a r > row M: r^(-(M-1)) r^(-(M-2)) ... r^(-1) a where multplication of the elements of the ith row and jth column and > jth row and ith column is always 1 (a_{ij}a_{ji} = 1) and a_{ii} = a > The determinant is > (a-1)^(M-1)(a+M-1) === Subject: Re: Epistemology 202: Advanced Topics >Answer this questions then: >What is an irrational number? >> A ratio which can be pointed out on straight line segments using only >> straight line segments and right angles which cannot be pointed out on >> straight line segments using bisection. >Where do you get this definition from? >> My mind mostly. >What kind of number is the seventh root of five? >> A transcendental ratio which cannot be pointed out except by using the >> intersection between straight line segments and a curve. >Where do you get this definition from? >> My mind mostly. >So let me get this straight. You use definitions for common concepts >which have little if anything in common with the usual terminology, >both in the specific field and in English as commonly used, and want >to base your universal truth system on this? No I want to base it on finite tautological regression to self contradictory alternatives. >And on what basis does your mind create these definitions? Finite tautological regression to self contradictory alternatives. === Subject: Re: Epistemology 202: Advanced Topics >> > Is geometry not a subset of logic? (ehehe he said set ehehe) Logic is a tool used in geometry. This doesn't make either a subset of >the other. >> >> So geometry and logic are subsets of something illogical? >(ehehe he said set again ehehe) >As they exist in your mind, yes. >> And as they exist in your mind? >As they exist in my mind, logic is a tool used in geometry. A subset of the illogical. === Subject: Re: Epistemology 202: Advanced Topics >Not really. The not is the *distinction* between them. Each is a >*negation* of the other. This is not contradictory. >> It's not self contradictory. It is contradictory or they would be the >> same. >Your definition of contradiction, again? >> One thing not another. >That's not what contradiction means in English. It isn't. Coulda fooled me, Tolstoij. Let's try again: If P not P is self contradictory and P is self then not is contradictory. === Subject: Re: Epistemology 202: Advanced Topics >> [...] >> I'm familiar with universal truth and my terminology is more than >> adequate to address things universally true whether or not that >> includes your understanding of mathematics being more difficult to >> say. > If you want to replace modern math, wouldn't it help if your work > addresses the same concepts? >> Will, Lester believes in universal truth, and he also believes he can >> find it just by thinking about it. It keeps him happy. >That much I've figured out... I'm just trying to figure out how lost a >cause he is. Right now it appears that he is unwilling to engage in >discussions at a technical level, even though he proposes revamping >technical subject matter. You don't consider contradiction, finite tautological regression, angular momentum, Planck's constant, hermit functions and SR technical? My mistake. === Subject: Re: Epistemology 202: Advanced Topics >Nope. The intersection of a line is nothing more than where the different >lines are the same. If there is more than one point that is the same >between two lines they do not intersect, as what it must mean is that one >line is contained in the other or they are the same line. > Allan, curves can intersect lines at more than one point. Straight > lines cannot intersect other straight lines at more than one point. True , but not a big deal really. We can still analyze those points differently because they are different in location on the line itself. === Subject: Re: Epistemology 202: Advanced Topics > What is NOT going to happen to to deduce a complete empirically correct > theory a prior from some logical principle. In order to come up with an > emprically correct theory of physical reality (which is reality) one > must do at least one measurement. We can't pull reality out of our heads. You mean at least one experience. We don't actually have to measure anything to get the theory. === Subject: Re: Epistemology 202: Advanced Topics > You mean at least one experience. We don't actually have to measure > anything to get the theory. No. I mean measure (by experimental means). Physics is a quantitative science, not a qualitative science. If it ain't quantitative it is nonsense. For example, the Standard Model has about 20 parameters all of which have to measured empirically. They cannot be deduced. The holy grail of a background free (which means a priori) theory is a futile quest. Just because God thought up the universe in His head does not mean that we can. Bob Kolker === Subject: Re: Epistemology 202: Advanced Topics > They are contradictions in space because they are not space, but are things > IN space. But when you draw a line in space you turn space into not-space. > Thus, a contradiction in space (to a certain point of view). > Did Lester bite you? You are beginning to sound like him. Contradiction > is a logical term, not a geometric term. Drawing a line in space does > not turn anything into anything else. A line in space is a subset of > points in space defined by certain conditions. A straight line in the > two dimensional space R^2 is a set of points (x,y) satisfying a*x + b*y > = c where a, b, c are given real constants such that a^2 + b^2 != 0. > That is a line (i.e. a straight line). Where is the contradiction? > If by line you mean a curve, that is different. We know of fractal > curves that can fill all of n space for any finite n. So let me get this straight: a line is nothing more than a set of locations and can never, ever fill space, and is only ever a part of space. But a curve -- which is not appreciably different in a strict geometric sense -- CAN fill space and be a thing that is not space. Why isn't Stephen harping on you about this statement, since it is contradictory? After all, why should curves fill space and lines not? === Subject: Re: Epistemology 202: Advanced Topics > So let me get this straight: a line is nothing more than a set of locations > and can never, ever fill space, and is only ever a part of space. But a > curve -- which is not appreciably different in a strict geometric sense -- > CAN fill space and be a thing that is not space. No. Sraight line cannot fill a plane or higher dimeinsional euclidean space. However curves can. Google . Your intuition that a curve is some kind of a bent or slightly bent straight line is incorrect, as Peano showed. The Peano curve is one of an infinite class of fractal curves, and they are very, very counter intuitive. Which shows that intuition is not always a reliable thing. Logic is. The mathematicians at the time, had a fit when Peano showed how to contruct his continous curve which filled a unit square. They had a similar fit when Wierstrass defined a continuous function which is no-where differentiable. The intuitive notion of literally drawing a curve carries with it physical motion which is smooth excepts at a countable number of points. This intuition simply is not correct, although it is seductive. Common sense and intuition are sometimes snares and delusions. Bob Kolker === Subject: Re: Epistemology 202: Advanced Topics [...] >Thus, a contradiction in space (to a certain point of view). >>Did Lester bite you? You are beginning to sound like him. Contradiction >>is a logical term, not a geometric term. Drawing a line in space does >>not turn anything into anything else. A line in space is a subset of >>points in space defined by certain conditions. A straight line in the >>two dimensional space R^2 is a set of points (x,y) satisfying a*x + b*y >>= c where a, b, c are given real constants such that a^2 + b^2 != 0. >>That is a line (i.e. a straight line). Where is the contradiction? >>If by line you mean a curve, that is different. We know of fractal >>curves that can fill all of n space for any finite n. > So let me get this straight: a line is nothing more than a set of locations > and can never, ever fill space, and is only ever a part of space. But a > curve -- which is not appreciably different in a strict geometric sense -- > CAN fill space and be a thing that is not space. A curve may include all the points in n-space, while a straight line includes only a subset of the points in n-space. In any n-space | n >=2, there is only one straight line between any two points, but infinitely many curves. (That is the axiomatic defintiton of a straight line as opposed to a curve.) Curves and straight lines are actually appreciably different. Very different. > Why isn't Stephen harping on you about this statement, since it is > contradictory? After all, why should curves fill space and lines not? Line == curve is not the same as line == straight line. === Subject: Re: Epistemology 202: Advanced Topics > They are contradictions in space because they are not space, but are things > IN space. > First of all, Lester rejects these set-centric views, so talking about > in here is quite out of his scope. You say about 5 posts before Lester agrees with my phrasing ... I fail to see how talking about in is a set-centric view. If I put something in a box, that is not a set-centric idea, but a physical-centric idea. I need no notion of sets to talk about doing so. Secondly, I really don't see > what's so contradicting in something being in something else but not > being that something else. I mean, the keyboard is in my laptop but > it's not my laptop. And the keyboard does not contradict it. Actually, this is a particular issue. The comment is that your keyboard is, indeed, your laptop (or a part of it). If you claim that lines are parts of space, then you can make the same analogy. But when _I_ talked about space, I meant empty space. And it seems odd to me to claim that a region of space that has a line drawn on it is still (empty) space. It's the line, and not empty anymore. And so a line is not empty space, by my reasoning, and so not part of empty space at all. It exists as a thing (not merely physical as Stephen would suggest, but independent of realm) inside the space that is defined and empty around it. > But when you draw a line in space you turn space into not-space. > Uh? How would drawing a line in space alter the space? Explained above. Your view only works if you consider a drawn line to be a part of space, or still space. === Subject: Re: Epistemology 202: Advanced Topics [...] But when _I_ talked about > space, I meant empty space. Then you should have said so. Empty space is a tricky concept, since, for example, it's not clear how one would describe its extent - or even if the notion of the extent of empty space makes sense. IOW whether it makes sense to speak of the dimensions of an empty space. > And it seems odd to me to claim that a region > of space that has a line drawn on it is still (empty) space. It's the line, > and not empty anymore. Granted. I would go a step further: without the object, there is no space. See my comments above. (line means straight line in the following comments.) A line is 1D space. Whether the line exists in an N-D space is another question. It's clear that lines in 2D flat space must either be parallel or intersect. That is not true of lines in an N-D space | N>2. So I would say not that a line makes a space non-empty, but that a line creates a space. Mathematically: any axiom about lines implies a space, namely the space that is the line itself, and possibly (depending on the axioms) a space of greater dimensionality than 1 in which the line is embedded. But embed here is itself a notion whose meaning depends on the axioms - see Lester's confusion about great circles on spheres, which he conceives as embedded in a 3D space, and so denies that it makes sense to speak of them as straight lines in (or on) the surface of the sphere. > And so a line is not empty space, by my reasoning, > and so not part of empty space at all. It exists as a thing (not merely > physical as Stephen would suggest, but independent of realm) inside the > space that is defined and empty around it. There need be no space around the line. It all depends on the axiom set athat you are using. >But when you draw a line in space you turn space into not-space. >>Uh? How would drawing a line in space alter the space? > Explained above. Your view only works if you consider a drawn line to be a > part of space, or still space. You are interpreting Lester's comment as A line creates a space or A line adds something to empty space. That's a guess, since Lester is nothing if not obscure. My guess is that Lester is saying that a line defines a minimal 1D space. But his other comments about lines, space, points and such indicate he hasn't understood the implications of what he's saying - if indeed he intends the meaning as I guessed it. === Subject: Re: Epistemology 202: Advanced Topics >[...] But when _I_ talked about >> space, I meant empty space. >Then you should have said so. Empty space is a tricky concept, since, >for example, it's not clear how one would describe its extent - or even >if the notion of the extent of empty space makes sense. IOW whether it >makes sense to speak of the dimensions of an empty space. >> And it seems odd to me to claim that a region >> of space that has a line drawn on it is still (empty) space. It's the line, >> and not empty anymore. >Granted. I would go a step further: without the object, there is no >space. See my comments above. (line means straight line in the >following comments.) A line is 1D space. Whether the line exists in an >N-D space is another question. It's clear that lines in 2D flat space >must either be parallel or intersect. That is not true of lines in an >N-D space | N>2. So I would say not that a line makes a space non-empty, >but that a line creates a space. Mathematically: any axiom about lines >implies a space, namely the space that is the line itself, and possibly >(depending on the axioms) a space of greater dimensionality than 1 in >which the line is embedded. But embed here is itself a notion whose >meaning depends on the axioms - see Lester's confusion about great >circles on spheres, which he conceives as embedded in a 3D space, and so >denies that it makes sense to speak of them as straight lines in (or on) >the surface of the sphere. If a straight line is one dimensional then a curve is 2+ dimensional. If a plane is 2 dimensional then a curved surface is 3+ dimensional. Unless Wolf wishes to claim that straight lines are zero dimensional or planes one dimensional, of course, there is no confusion except in Wolf's rather lurid imagination. >> And so a line is not empty space, by my reasoning, >> and so not part of empty space at all. It exists as a thing (not merely >> physical as Stephen would suggest, but independent of realm) inside the >> space that is defined and empty around it. >There need be no space around the line. It all depends on the axiom >set athat you are using. >>But when you draw a line in space you turn space into not-space. >Uh? How would drawing a line in space alter the space? >> Explained above. Your view only works if you consider a drawn line to be a >> part of space, or still space. >You are interpreting Lester's comment as A line creates a space or A >line adds something to empty space. That's a guess, since Lester is >nothing if not obscure. My guess is that Lester is saying that a line >defines a minimal 1D space. But his other comments about lines, space, >points and such indicate he hasn't understood the implications of what >he's saying - if indeed he intends the meaning as I guessed it. === Subject: Re: Epistemology 202: Advanced Topics > [...] But when _I_ talked about >> space, I meant empty space. > Then you should have said so. Empty space is a tricky concept, since, > for example, it's not clear how one would describe its extent - or even > if the notion of the extent of empty space makes sense. IOW whether it > makes sense to speak of the dimensions of an empty space. Mathematically there is no such thing as empty space. Space consists of abstract objects. The only truly empty mathematical space is the empty set and nobody does abstract geometry on empty sets. Empty space in the physical sense is space devoid of matter. If one counts fields and energy as contents then there are no physically emptry spaces either. Bob Kolker === Subject: Re: Epistemology 202: Advanced Topics >> [...] But when _I_ talked about > space, I meant empty space. >> Then you should have said so. Empty space is a tricky concept, >> since, for example, it's not clear how one would describe its extent - >> or even if the notion of the extent of empty space makes sense. IOW >> whether it makes sense to speak of the dimensions of an empty space. > Mathematically there is no such thing as empty space. Space consists > of abstract objects. The only truly empty mathematical space is the > empty set and nobody does abstract geometry on empty sets. > Empty space in the physical sense is space devoid of matter. If one > counts fields and energy as contents then there are no physically emptry > spaces either. > Bob Kolker Yup, that's what I was hoping ACC would intuit when he read my comment. === Subject: Re: Epistemology 202: Advanced Topics >> [...] But when _I_ talked about > space, I meant empty space. >> Then you should have said so. Empty space is a tricky concept, since, >> for example, it's not clear how one would describe its extent - or even >> if the notion of the extent of empty space makes sense. IOW whether it >> makes sense to speak of the dimensions of an empty space. >Mathematically there is no such thing as empty space. Space consists >of abstract objects. The only truly empty mathematical space is the >empty set and nobody does abstract geometry on empty sets. >Empty space in the physical sense is space devoid of matter. If one >counts fields and energy as contents then there are no physically emptry >spaces either. Except between your ears. === Subject: Re: Epistemology 202: Advanced Topics > If you want to replace modern math, wouldn't it help if your work > addresses the same concepts? >> I would be interested in a summary of what you consider the concepts >> of modern math. > Set theory, mathematical logic, abstract algebra, real analysis, > topology, geometry, statistics, probability, number theory, > combinatorics, graph theory, and more. Ah, I see. You don't know what 'concept' means. -- For at least another hundred years we must pretend to ourselves and to every one that fair is foul and foul is fair; for foul is useful and fair is not. Avarice and usury and precaution must be our gods for a little longer still. --John Maynard Keynes === Subject: Re: Epistemology 202: Advanced Topics >> If you want to replace modern math, wouldn't it help if your work >> addresses the same concepts? > I would be interested in a summary of what you consider the concepts > of modern math. >> Set theory, mathematical logic, abstract algebra, real analysis, >> topology, geometry, statistics, probability, number theory, >> combinatorics, graph theory, and more. > Ah, I see. You don't know what 'concept' means. Ok, sets, logic, groups, rings, fields, metric spaces, relationships between points and lines, data summarization and prediction, properties of integers, methods of counting, etc. Then again, do you think this will help Lester any? -- Will Twentyman email: wtwentyman at copper dot net === Subject: Re: Epistemology 202: Advanced Topics > If you want to replace modern math, wouldn't it help if your work > addresses the same concepts? > I would be interested in a summary of what you consider the concepts >> of modern math. > Set theory, mathematical logic, abstract algebra, real analysis, > topology, geometry, statistics, probability, number theory, > combinatorics, graph theory, and more. >> Ah, I see. You don't know what 'concept' means. > Ok, sets, logic, groups, rings, fields, metric spaces, relationships > between points and lines, data summarization and prediction, properties > of integers, methods of counting, etc. > Then again, do you think this will help Lester any? Don't know. That wasn't why I asked. I suppose I was curious what you thought a 'concept' was and if and how you would distinguish between a 'concept' and a 'system'. -- For at least another hundred years we must pretend to ourselves and to every one that fair is foul and foul is fair; for foul is useful and fair is not. Avarice and usury and precaution must be our gods for a little longer still. --John Maynard Keynes === Subject: Re: Epistemology 202: Advanced Topics > Don't know. That wasn't why I asked. I suppose I was curious what you > thought a 'concept' was and if and how you would distinguish between a > 'concept' and a 'system'. A concept is a basic idea and he listed several of many that are part of mathematical theories or systems. I alread discussed one with you, the concept of a function or mapping. That is very basic and it depends on set theory. Bob Kolker === Subject: Re: Epistemology 202: Advanced Topics >> Don't know. That wasn't why I asked. I suppose I was curious what you >> thought a 'concept' was and if and how you would distinguish between a >> 'concept' and a 'system'. > A concept is a basic idea and he listed several of many that are part of > mathematical theories or systems. I alread discussed one with you, the > concept of a function or mapping. That is very basic and it depends on > set theory. Wrong, wrong and wrong. -- There are many things for which mathematical modeling leads at best to fuzzy, contingent, statistical results and never successfully predicts 'new entities' at all. In fact, such systems are the rule, not the exception. So the proper answer to the question Why is mathematics so marvelously applicable to my science? is simply Because that's the kind of science you've chosen to study! http://www.catb.org/~esr/writings/utility-of-math/ === Subject: Re: Epistemology 202: Advanced Topics > Then again, do you think this will help Lester any? Only a brain transplant can help Lester. Bob Kolker === Subject: Re: Epistemology 202: Advanced Topics >> Then again, do you think this will help Lester any? >Only a brain transplant can help Lester. Just as long as it isn't yours. === Subject: Re: Epistemology 202: Advanced Topics > Ah, I see. You don't know what 'concept' means. The concepts of modern mathematics are the abstract objects that modern mathematics deals with or operates upon. There are literally thousands of conepts that mathematics develops and uses. One of the most basic concepts dealt with in mathematics is the concept of a function (aka mapping). A function is a subset of a cartesian product, hence a relation with some additional restrictions. In order to define a function one must first have the concept of set. Virtually evrey branch of mathematics that can be applied to physics deals with functions which in turn require sets. Bob Kolker === Subject: Re: Epistemology 202: Advanced Topics > Ah, I see. You don't know what 'concept' means. > The concepts of modern mathematics are the abstract objects that modern > mathematics deals with or operates upon. There are literally thousands > of conepts that mathematics develops and uses. One of the most basic > concepts dealt with in mathematics is the concept of a function (aka > mapping). Ahem. Mapping IS a function, but there is no guarantee that all functions are mapping or work out precisely as mapping does in set theory. In short, you do mappings using a mapping function, but that does not make all functions set-theory mappings. If my intent is not to deal with sets, then my functions do not do set-theory mapping. You have elevated set-theory to the extent that you believe that if any functionality is used in sets then it is sets that all uses of that functionality aims at. === Subject: Re: Epistemology 202: Advanced Topics > Ahem. Mapping IS a function, but there is no guarantee that all functions > are mapping or work out precisely as mapping does in set theory. A function is a relation with restrictions guaranteeing single valuedness. The term mapping and function are virtually synonymous. A mapping is a rule that associates to each element in the domain of the mapping an element in the co-domain. Take your ahem and stick it up your a posteriori. Bob Kolker === Subject: Re: Epistemology 202: Advanced Topics >> Ah, I see. You don't know what 'concept' means. > The concepts of modern mathematics are the abstract objects that modern > mathematics deals with or operates upon. There are literally thousands > of conepts that mathematics develops and uses. One of the most basic > concepts dealt with in mathematics is the concept of a function (aka > mapping). A function is a subset of a cartesian product, hence a > relation with some additional restrictions. In order to define a > function one must first have the concept of set. Virtually evrey branch > of mathematics that can be applied to physics deals with functions which > in turn require sets. But not infinite sets nor different kinds of infinity. Sets are also often misused. -- For at least another hundred years we must pretend to ourselves and to every one that fair is foul and foul is fair; for foul is useful and fair is not. Avarice and usury and precaution must be our gods for a little longer still. --John Maynard Keynes === Subject: Re: Epistemology 202: Advanced Topics > But not infinite sets nor different kinds of infinity. Sets are also > often misused. Wrong. Infinite sets with the cardinality of the real numbers form the domains of the functions used in physics. Countable sets index the sequence and the series that are used to approximate the functions. Full bore ordinal number theory is not used in physical applications as far as I know. Bob Kolker === Subject: Re: Epistemology 202: Advanced Topics Is geometry not a subset of logic? (ehehe he said set ehehe) Logic is a tool used in geometry. This doesn't make either a subset of >the other. >>So geometry and logic are subsets of something illogical? >(ehehe he said set again ehehe) >As they exist in your mind, yes. >>And as they exist in your mind? > As they exist in my mind, logic is a tool used in geometry. Logic is a tool used by the human mind in almost every encounter with the world. But, I suppose, to a man with a hammer... -- For at least another hundred years we must pretend to ourselves and to every one that fair is foul and foul is fair; for foul is useful and fair is not. Avarice and usury and precaution must be our gods for a little longer still. --John Maynard Keynes === Subject: Re: Epistemology 202: Advanced Topics > Logic is a tool used by the human mind in almost every encounter > with the world. You are saying Lester is not human? -- Giuseppe Oblomov Bilotta Axiom I of the Giuseppe Bilotta theory of IT: Anything is better than MS === Subject: Re: Epistemology 202: Advanced Topics >>Logic is a tool used by the human mind in almost every encounter >>with the world. > You are saying Lester is not human? No. I'm saying that you are full of . -- There are many things for which mathematical modeling leads at best to fuzzy, contingent, statistical results and never successfully predicts 'new entities' at all. In fact, such systems are the rule, not the exception. So the proper answer to the question Why is mathematics so marvelously applicable to my science? is simply Because that's the kind of science you've chosen to study! http://www.catb.org/~esr/writings/utility-of-math/ === Subject: Re: Epistemology 202: Advanced Topics > Logic is a tool used by the human mind in almost every encounter with > the world. But, I suppose, to a man with a hammer... To a man with a hammer Lester Zick's head looks as pointed as a nail. Bob Kolker === Subject: Re: Epistemology 202: Advanced Topics >> Logic is a tool used by the human mind in almost every encounter with >> the world. But, I suppose, to a man with a hammer... >To a man with a hammer Lester Zick's head looks as pointed as a nail. Curiously spoke indeed for one who can't point out much of anything. === Subject: Re: Epistemology 202: Advanced Topics >Not really. The not is the *distinction* between them. Each is a >*negation* of the other. This is not contradictory. >>It's not self contradictory. It is contradictory or they would be the >>same. >Your definition of contradiction, again? >>One thing not another. > That's not what contradiction means in English. What does 'transcendent' mean in English? > Is there a Lesterish-English dictionary somewhere? -- For at least another hundred years we must pretend to ourselves and to every one that fair is foul and foul is fair; for foul is useful and fair is not. Avarice and usury and precaution must be our gods for a little longer still. --John Maynard Keynes === Subject: Re: Epistemology 202: Advanced Topics > What does 'transcendent' mean in English? http://www.m-w.com/cgi-bin/dictionary?va=transcendent Hey, Albert, wasn't that you the one that complained about mathematicians using terms in ways which couldn't be found in dictionaries? How comes you take that from Lester? -- Giuseppe Oblomov Bilotta They that can give up essential liberty to obtain a little temporary safety deserve neither liberty nor safety. Benjamin Franklin === Subject: Re: Epistemology 202: Advanced Topics >>What does 'transcendent' mean in English? > http://www.m-w.com/cgi-bin/dictionary?va=transcendent > Hey, Albert, wasn't that you the one that complained about > mathematicians using terms in ways which couldn't be found in > dictionaries? How comes you take that from Lester? I'm taking nothing from Lester. I'm illustrating your hypocrisy to the world. -- There are many things for which mathematical modeling leads at best to fuzzy, contingent, statistical results and never successfully predicts 'new entities' at all. In fact, such systems are the rule, not the exception. So the proper answer to the question Why is mathematics so marvelously applicable to my science? is simply Because that's the kind of science you've chosen to study! http://www.catb.org/~esr/writings/utility-of-math/ === Subject: Re: Epistemology 202: Advanced Topics > What does 'transcendent' mean in English? In general parlance transcendent means beyond the usual. In mathematical parlance transcendental (applied to real numbers) means non-algebraic real numbers. In a sence a non-algebraic real number is beyond the reach of ordinary polynomial functions. I suspect this is how the usage came about a long time ago. Bob Kolker === Subject: Re: Epistemology 202: Advanced Topics >> What does 'transcendent' mean in English? >In general parlance transcendent means beyond the usual. And curves are definitely beyond the usual straight line. === Subject: Re: Epistemology 202: Advanced Topics >> What does 'transcendent' mean in English? > In general parlance transcendent means beyond the usual. In mathematical > parlance transcendental (applied to real numbers) means non-algebraic > real numbers. In a sence a non-algebraic real number is beyond the reach > of ordinary polynomial functions. I suspect this is how the usage came > about a long time ago. who now utilizes the arguments that just weeks ago he complained about vociferously when used by others. Hypocrisy. -- For at least another hundred years we must pretend to ourselves and to every one that fair is foul and foul is fair; for foul is useful and fair is not. Avarice and usury and precaution must be our gods for a little longer still. --John Maynard Keynes === Subject: Re: Epistemology 202: Advanced Topics >>The keyboard is certainly not your computer. It doesn't contradict >>your computer in general but does contradict other parts such as the >>display. >It doesn't. Indeed, my laptop has both a keyboard and a display. They >don't contradict each other, they coexist. >>Yet they are different. Each is not the other. The not is a >>contradiction between them. > Not really. The not is the *distinction* between them. Each is a > *negation* of the other. So, a right and a left hand equal zero hands? This is not contradictory. > Your definition of contradiction, again? -- For at least another hundred years we must pretend to ourselves and to every one that fair is foul and foul is fair; for foul is useful and fair is not. Avarice and usury and precaution must be our gods for a little longer still. --John Maynard Keynes === Subject: Re: Epistemology 202: Advanced Topics > So, a right and a left hand equal zero hands? Can you have them occupy the same space? -- Giuseppe Oblomov Bilotta E la storia dell'umanit.88, babbo? Ma niente: prima si fanno delle cazzate, poi si studia che cazzate si sono fatte (Altan) (And what about the history of the human race, dad? Oh, nothing special: first they make some foolish things, then you study what foolish things have been made) === Subject: Re: Epistemology 202: Advanced Topics