mm-3519 === Subject: Integration Can anyone help me out with this. I've tried and failed a number of ways. Int[ 1 / ( 1 + ( (a/b)*Tan[x] )^2 ) ] dx I can get methematica to give me the answer which is: ( b^2*x - a*ArcTan[ (a/b)*Tan[x] ] ) / (b^2-a^2) But try as I might I can't see analytivcally why this is so. I understand the presense in the answer of the term ArcTan[ (a/ b)*Tan[x] ] as the original integral is of the form 1/(x^2 + k^2) where k = 1 and x = (a/b)*Tan[x]. I've tried some substitutions for x but with no luck. Mitch. === Subject: Re: Integration _Any_ rational function of sines and cosines (which is what you have) can be integrated by the substitution t = tan(x/2) whence tan(x) = 2t/(1 - t^2) dx = 2 dt/(1 + t^2). That will give a rational function in t to be integrated dt. Rational functions P(t)/Q(t) with degree P < degree Q can be integrated by first. I don't claim that this is the quickest way to do your integral but it is generally applicable. -- Remove antispam and .invalid for e-mail address. We have lingered in the chambers of the sea By sea-girls wreathed with seaweed red and brown Till human voices wake us, and we drown. === Subject: Re: Integration On 26 Feb, 21:11, Frederick Williams post almost as soon as I had sent it because it dawned on me how I could do it. I had hoped I'd deleted it before anyone would have read it. It shows just how helpful and efficient this group is as I deleted it within a couple of minutes! This is what I did: Make the substitution z = (a/b)*Tan[x] so that dx = ((b/a)*(cos[x])^2)dz (1) and use 1/(cos(x))^2 = 1 + (tan[x])^2 to substitute for (cos[x])^2 in (1) finally giving: dx = (ab/(a^2+b^2z^2))dz So the integral becomes: ab*(a^2-b^2)*Int[ 1/( ( a^2+b^2*z^2 ) * ( 1+z^2 ) ) ] dz And this I do by splitting the integrand into partial fractions so leaving me to evaluate: ab*Int[1/(1+z^2)]dz - ab^3*Int[1/(a^2+b^2z^2)]dz = abArcTan[z] - b^2ArcTan[bz/a] Which after substitutiong back for z agrees with Mathematica's asnwer of: ab*ArcTan[(a/b)*Tan[x]] - b^2*x Mitch. === Subject: Re: Integration You lost a denominator of (a^2 - b^2) somewhere along the way. Furthermore, as I pointed out earlier, even with the correct denominator, your answer is _not_ the same as Mathematica's. But that's not too important because both your answer and Mathematica's have needless jump discontinuities, and thus neither is an antiderivative valid over the whole real line. Suppose that a and b are positive and unequal. Then an antiderivative, continuous over the reals, is b x/(b + a) + a b/(b^2 - a^2) ArcTan[(b^2 - a^2)Sin[2x]/((b^2 - a^2)Cos[2x] + (b + a)^2)] David W. Cantrell === Subject: Re: Integration Also when a = b the integrand has a different form (repeated irreducible quadratic instead of product of unrepeated irreducible quadratics) and needs separate attention. --Lynn === Subject: Re: Integration BTW, in case anyone has forgotten, this thread concerns getting an antiderivative for 1/(1 + ((a/b)*Tan[x])^2) with respect to x. Note that I had specified that a and b were unequal: But to deal with the case when a = b, we need not necessarily do a new integration. We can instead simply take the limit of the result above, say, In[5]:= Limit[b x/(b + a) + a b/(b^2 - a^2) ArcTan[(b^2 - a^2)Sin[2x]/((b^2 - Out[5]= (1/2)*(x + Cos[x]*Sin[x]) It should also be noted that if one attempts to get an antiderivative valid for a = b by this limiting process but using Mathematica's own generic antiderivative, the result will still contain needless jump discontinuities. David === Subject: Re: Integration Perhaps you meant to say u = (a/b)*Tan[x] above, as a substitution. That _does_ work. Maybe you just made a misstep somewhere and need to recheck your work. As to the result, I have some comments: 1. Surely you intended to type ( b^2*x - a*b*ArcTan[ (a/b)*Tan[x] ] ) / (b^2-a^2). Note the factor of b with the ArcTan, lacking in your expression above. 2. I'm suspicious. Did Mathematica really gave you an answer with the term b^2*x in it? I suspect instead that it gave you an answer with the term b^2*ArcTan[Tan[x]] in it. Note that ArcTan[Tan[x]] is _not_ identically x for all real x. 3. If you're wanting an antiderivative which is valid for all real x, then Mathematica's answer does not suffice, due to needless jump discontinuities. David W. Cantrell === Subject: Re: Integration Without the a/b complication, you have int(1 / ( 1 + tan^2(x) ) dx) let u = tan(x), du = sec^2(x) dx so dx = 1 / ( sec^2(x)) du = 1 / ( 1 + tan^2(x) ) du = 1 / (1 + u^2) du so your integral becomes int 1 / ( 1 + u^2)^2 du, repeated irreducible quadratic factor. Can you take it from there? --Lynn === Subject: Re: Differential Equations Is Occam's Razor disposable? If we assume that phenomena outside the powers of our observations do not exist, then we immediately cut away anything beyond the limits of our telescopes and smaller than the probing wavelengths of our (electron) microscopes. Moreover, we deny the existence of the transcendent, spirituality, etc. We assume none of these things exist, and then assert that within our system-of-thought, their existence is absurd! (Well, of course it is, if we take as an axiom their non-existence. You don't have to be Kurt Godel to see that!) Yes, as you say, we may use differential equations to model the Universe, as in applied maths, but any mathematical model is based upon approximations to the ultimate reality (if indeed there _is_ an ultimate reality). Our best scientific theories are only approximations to what is really going on in the objective world. We never completely grasp the _true_ laws of physics (if indeed there _is_ an ultimate theory). Any scientific theory may ultimately be overturned and replaced by a closer approximation to reality, the classic example being the overhaul of Newtonian gravitation (c. 1670) by Einsteinian general relativity (c. 1915). Relativity was more accurate, especially in its predictions about the gradual precession of Mercury's orbit around the Sun. Yes, the aim of science is to give a broader understanding of the laws of the Universe. We make an assumption that the Universe works according to logic. But surely this must be the case: Logic is just the means of deducing further theorems from a given set of axioms. If the axioms are true, then the theorems cannot fail to be true. Surely, these ideas must hold sway in the physical Universe just as in the austere world of pure-mathematical formal systems! For example: Suppose the spatial Universe really did adhere to the axioms of Euclidean geometry. In that case, any actual physical triangle's three angles could not fail to add up to 180 degrees. Likewise, any right-angled triangle could not fail to adhere to Pythagoras's theorem. But there is always the slight fearful possibility that rationalism is not all it is cracked up to be. In other words, the Universe itself may be wilder and crazier than the more smart, polite, well-dressed, old-fashioned Rationalism. It may (in some parts and at some times) behave illogically, unmathematically, irrationally and chaotically. There may exist, in objective reality, some regions of the Cosmos that exist outside the boundaries of Rationalism! It's a frightening thought to contemplate... === Subject: Re: Differential Equations Just a few comments from the peanut gallery: Aside from the gobbledygook - where is(are) the differential equation(s) in the mess that is the above paragraph? The whole is devoid of meaning... Have you other evidence for the existence of things that cannot be (even in principle) observed? How should we logically evaluate that evidence? Obviously you do not deny their existence - why? Have you observed their existence or effects (which would imply their existence)? In no- one has ever observed them, where did the terms come from? I don't know about others, but I neither assume their existence nor their nonexistence, as neither question is pertinent to those questions for which I use logic do determine the answers. I'm not sure (it's difficult to decipher that first paragraph), but I don't think it actually said anything about modelling the universe (nor even a part of the universe). Of course- differential equations are commonly used to model many physical phenomena, and - as you point out - they are approximations to what the system 'actually' does (or, as you say, 'approximations to the ultimate reality'). You say that like it's a bad thing. As though there is some (non- scientific) way to know what *is* really going on. So- what is this way you are talking around without actually mentioning explicitly? Yes- so? Within the limits of the observations available at the time, Newton's theory was as good as could be got. I would point out that Newton's laws of motion have still not been 'overturned'. Could it be that, in that case, we actually *do* know 'what is really going on'? But- even so, I am willing to accept the idea that ever finer observation may lead to a refinement ('overhaul' sounds like the two theories have nothing to do with one another- yet Newton's theory of gravitation is Einstein's theory taken in the limit of slow speeds and low masses.) of Newton's laws of motion. Would that make his laws of motion *wrong*? Working in a lab, I have gotten the impression that the aim of science is to secure funding. Broader understanding - meaning what exactly? Yes- have you an example where the universe fails to work according to logic? A repeatable experiment I could perform? A very specific and limited definition of logic. True- but logic is also an analytical tool which can be used to examine and infer from experimental data to determine not only if your theory is 'true', but also in what ways it might fail, allowing refinement of your initial set of assumptions... [Within the limits of accuracy of your system of measurement - for example, if your system of measurement includes students who have never measured an angle with a protractor before, you would see that the three angles of a triangle can sum to many interesting values]. I would ask also what you mean by an 'actual physical triangle'? But such a fear is irrational, and so I, being rational, do not feel it. What color are Rationalism's pants? I haven't met Rationalism, I didn't realize he was so polite. Please don't anthropomorphise if you are trying to make some kind of point. At what parts? When? Many things 'may' be, and it is easy to assert the possibility- what is your reasoning that we should take into account such a possibility while working on our 'old-fashioned' rational ideas? If it is in objective reality, then it is observable- while that does not imply the observations (or the universe, if you prefer) will be rational, I will deal with the lack of rationality only after I have observed it. Until then, it is not frightening to contemplate, rather it is pointless. Tell me- what will you do, upon observing this irrational event. How would you even recognize it? What would a violation of 'rationalism' look like? I take two marbles and push them next to two others and all of a sudden there are five? I draw a triangle on a smooth, flat surface and measure its angles and notice that they add to 314 degrees? (And even *that* doesn't make the universe irrational, merely non-Euclidean). Finally- what does this have to do with mathematics? Does the validity of mathematical theory depend upon physical reality? If so, how? cheers- Eric === Subject: Re: Differential Equations I think I have upset you, and for this I'm sorry. My aim is to get talking about Seung Kim's idea and hopefully, to get Seung Kim to explain them a bit more clearly. I am just throwing a few ideas into the debate. I don't want to attack anyone's worldview, just to make a few suggestions. No. How can there be empirical evidence for things that are not in principle observable? They may well exist. I have not observed them in a scientific way, but there have been times when I have experienced subjective emotions like love, that can't be dealt with scientifically. These terms came from our own ideas, experiences, theories... A good position to take. Wait for the clinching proof before you make up your mind which way to believe. But the clinching proof may never come, perhaps because it _can_ never come. Not all theorems can be proved. Well, Seung Kim (if I understand him/her correctly) was talking about differential equations and how they lead to a broader understanding of the laws in the universe. In my opinion, differential equations are used to model parts of the universe, eg, the Navier-Stokes equations model the atmosphere and could be used for qeather forecasting. As for the Universe as a whole, look at cosmology and the FRW models, for instance. I don't think it's a bad thing. It's good that we have these theories, but they are only the best that we have at present. They are not perfect, and are subject to future improvement. To think that our current understanding of science covers everything is an oversimplification and unduly optimistic. This kind of thinking prevailed in the late 19th century, but the discovery of atomic physics, quantum mechanics and special relativity lead to an appreciation of greater complexity. I don't mention explicitly a way of knowing what is really going on, because there probably is none. I am just pointing out the limitations of our worldview. I completely agree. Perhaps so. Perhaps the 3 Laws of Motion are part of objective reality. Newton's theory and Einstein's theory are very different. The latter involves tensor calculus, curved space, differential geometry. They are based on different sets of assumptions. To give an example, Newton's second law, F=ma assumes that the mass of a given object is constant. In relativity, an object's mass increases as it approaches the speed of light. Well, I guess brutal economic facts can get in the way of idealism! Which is a shame. By broader understanding, I simply mean that we understand more about the Universe and understand it better and deeper, with a greater degree of complexity. No. But if it did behave like that, our experimental apparatus would behave illogically and so our experiment would be invalid. Thus, there is no logical experiment that could demonstrate the universe is illogical. Logic is such a fundamental assumption in science that we cannot test it using science. Not everything can be dealt with by repeatable experiments. So much of what matters is in the form of subjective experience, value-judgements, the senses, etc... Rationalism wants to exlude this, but life would be pretty dull if we got rid of love, beauty, humour, etc... Yes, there is more to logic than meets the eye. Perhaps I need a broader understanding. Here, the difference is between deduction (pure maths) and induction (science), the latter being not completely logical. Yes! Human error can be a problem! One that you built in the real physical world, out of wood or something. The problem is: its edges would not be completely straight. Most people have irrational fears. It's just part of life, like other emotions. Grey. Oh, you should meet him! He's ever such a nice chap! But he can be a bit dull sometimes. Sorry, Eric. Regions of undiscovered space. Black holes. Other universes. It's a real possibility and it could be happening somewhere out there. Well, maybe we can't do both things at the same time. We think differently in different situations. On weekdays, we carry out logical experiments. On Sunday afternoons, we speculate on the possibilities of irrationalism. I don't know, but it may not just be a case of observation. It may be a case of active involvement. To fall in love is to create a religion whose god is fallible. I don't know what a violation of rationalism would look like, but that's a very interesting question. If you only use empirical observations, you will only gain access to those parts of the universe that are accessible via empricism. There may well be other parts, but we don't know what they are like. There is a temptation to dismiss these parts as mumbo-jumbo, but just because they are mumbo-jumbo doesn't mean they are not true. Sounds like the Banach-Tarski Theorem. Yes, I totally agree. No, it think it's vice versa. Any physical theory is based upon mathematical ideas. These ideas need to be valid. In a sense, pure maths come first, and then people take it any apply it to physics, or rather, they formulate physical theories in mathematical terms, the most famous way being the equation. However, there are huge swathes of pure maths that are not applied to physics at all. They are just there, and they will probably never be used for anything. But they are beautiful things, and it's worth having them. Our culture in immeasurably enriched by their presence. Well, Etic, we seem to have got into a bit of an argument. It reminds me of a kid I knew at school. We used to talk about stuff like this for hours, and he always took a rational standpoint. But I hope this leads to a friendship, and we could talk about something less controversial, like calculus or group theory. Do you have a favourite branch of maths? === Subject: Re: Differential Equations Argh! I just spent a good hour typing up a reply and it seems to have disappeared! So- David, I take no offense, and indeed apologize for my snarkiness (in replying I did notice that a couple of times I was a bit over the top!). You gave a well-reasoned response and I *tried* to do the same- but it seems to be lost in the non-observable aether... To summarize my ending statement: 1: shall we break this up into slightly more manageably bits and maybe take the philosophical parts over to alt.philosophy or some similar group? 2: for the math that we might enjoy discussing, I am currently taking an abstract algebra course (which I enjoy quite a bit) and am pretty happy to discuss any aspect of mathematics, though some bits I may have to resort to brute force and ignorance to get through! (I do have a basic background knowledge of topology, analysis, algebra, combinatorics and number theory - about what you'd expect from finishing a bachelor's in math from a state university. I know next to nothing about complex analysis, PDEs, numerical analysis, etc... Though I *am* interested!) Anyways- pick a topic and let's see where it goes. I have to do some other things right now, but I'll try to recreate my post later tonight or tomorrow. cheers! Eric Riley === Subject: Re: Differential Equations Sorry, I spelt you name wrong on the previous post. === Subject: Why is 2^prime -1 a prime? I've tried: 2^n - 1 for a lot of N. Prime N always makes a prime, nonprime N always makes a nonprime. It's interesting but before I start examining the factors that I get I'd like to ask for a simple explanation. Dan === Subject: Re: Why is 2^prime -1 a prime? Please define a lot of N. 2^p-1 is composite for p = 11,23,29,37,41,43,47,53,59,67,71,73,79,83,97,101,103,.... etc. etc. This assertion is false. (1) Please be careful about notation. n != N. (2) Although we still do not have a rigorous proof (there are various heuristics), 2^p - 1 will be composite for almost *all* prime numbers p. 2^p-1 will be prime *very* rarely. We do know that for p up to about 33 million, 2^p-1 is prime only 44 times. The ratio: #{p < N; 2^p-1 is prime}/#{p < N; p prime} goes to 0 as This is trivial high school algebra. Consider the polynomial x^(ab) - 1. It is divisible by x^a-1 and x^b-1. Now take x = 2. === Subject: Re: Why is 2^prime -1 a prime? === Subject: Re: Why is 2^prime -1 a prime? 2^11 - 1 = 2047 = 23 * 89. Part 1 of the simple explanation is that 2^(a*b) - 1 is always composite, because it has the algebraic factors 2^a - 1 and 2^b - 1, and usually more factors. Part 2 of the simple explanation is that if p is prime, and 2^p - 1 is composite, its factors MUST be of the form n*p + 1 (as in 2^11 - 1, above). This is a consequence of Fermat's Little Theorem. For smallish p, there are seldom enough possible factors. -- === Subject: MI5 Persecution: Stand up for Free Speech 14/8/95 (1175) [snip a large pile of winging complaining drivel] Geez what a bunch of tossers you all are - you don't like someones postings so you try and get him evicted from the net, why not just use a kill file - you DONT have to read his posts/threads now do you ? Why is it the net is getting populated by the biggest bunch of self absorbed little Hitlers ? You don't like someones posts so you bloody complain or mail bomb them - grow up you bunch of ing sad gits ! Richard. 1175 === Subject: Re: MI5 Persecution: Stand up for Free Speech 14/8/95 (1175) Why IS the net getting populated by the biggest bunch of self absorbed little Hitlers? You don't like someones posts so you bloody complain and whine about what they're doing rather than employing your own killfile? Hypocritical much? === Subject: MI5 Persecution: Troubling Censorship Issues 20/8/95 (3830) : He posts this drivel every week or so to a number of groups that : I subscribe to and nothing seems to stop him. *ALL* of his posts : are off topic and unwelcome to the groups he posts to. : We have complained about him to his postmaster on at least four : previous occasions and he still posts the same crap. As his SP : seem unwilling, or unable, to do anything about him, we were : windering if there was anything you could do? If he is not actively using tactics to avoid or sabotage killfiles (posting from various systems, forging, massive crossposting to create cross-newsgroup flame wars) then it should be quite easy to killfile him. It does seem that the frequency and the size of his posts are approaching net abuse. However, IMHO, they aren't quite there yet. If his postmaster were to act in this instance, it would raise troubling censorship issues. -- Karen Lofstrom lofstrom@lava.net --------------------------------------------------------------------- Adventures for those of the inadmissable kind, with no follow through. --- a flaming bonzo beckwithian idiot 3830 === Subject: Adding decimals... answers is 33. I don't know what's wrong. http://img168.imageshack.us/img168/5186/damnitgz3.jpg === Subject: Re: Adding decimals... You seem to be working on the following calculation: 4.5 + 4.5 + 12 + 12 Try writing each 12 as 12.0 and then writing the calculation so that the decimal points line up, like this: 4.5 4.5 12.0 12.0 Add this up as usual, and it should come to 33.0, which equals 33. === Subject: Call for Papers with Extended Deadline: 2007 International Conference on Scientific Computing (CSC'07), June 25-28, 2007, USA Extended Paper Submission Deadline: March 4, 2007 The 2007 International Conference on Scientific Computing (CSC'07) Academic Sponsors: Research Labs at MIT, Harvard, Purdue, Univ. of Texas at Austin, ... URL: http://www.world-academy-of-science.org/worldcomp07/ws/CSC07 You are invited to submit a full paper for consideration. All accepted papers will be published in the conference proceedings. SCOPE of CSC'07: Topics of interest include, but are not limited to, the following: O Supercomputing and scientific computing O Mathematical modeling O Computational models O Computational electromagnetics O Computational electrodynamics O Computational fluid dynamics O Scientific visualization O Numerical methods and simulation O Partial differential equations O Monte Carlo methods and applications O Molecular dynamics O Stochastic differential equations O Optimization and optimal control O Ordinary differential equations O Finite element methods O Software architectures for scientific computing O Scientific computing and supercomputing benchmark design O Overlapping and nonoverlapping domain decomposition methods O Seismic Data Processing O Multi-level methods O Multi-grid methods O Iterative methods O Krylov methods O Level-set methods O Atmospheric science O Integral equations O Operational research O Dynamical systems O Generalized eigen-problems O Coupled problems O Nonsymmetric solvers O Nonlinear systems and eigenvalue solvers O Numerical linear algebra O Inversion problems in Geophysics O Approximation theory O Mathematics and circuit simulation O Mathematical software tools O Splines and wavelets and applications O Engineering problems and applications O Applications of scientific computing in physics, mechanics, chemistry, biology, environmental and hydrology problems, production scheduling, automotive industry, ... Submission of Papers: Prospective authors are invited to submit their full paper (about 5 to 8 pages - single space, font size of 10 to 12) to H. R. Arabnia by Mar. 4, 2007 (hra@cs.uga.edu). E-mail submissions in MS Doc or PDF formats are preferable (postal mail submissions are also fine.) All reasonable typesetting formats are acceptable (later, the authors of accepted papers will be asked to follow a particular typesetting format to prepare their papers for publication.) The length of the Camera-Ready papers (if accepted) will be limited to 7 (IEEE style) pages. Papers must not have been previously published or currently submitted for publication elsewhere. The first page of the draft paper should include: title of the paper, name, affiliation, postal address, email address, and telephone number for each author. The first page should also identify the name of the author who will be presenting the paper (if accepted) and a maximum of 5 topical keywords that would best represent the content of the paper. Each paper will be refereed by two experts in the field who are independent of the conference program committee. The referees' evaluations will then be reviewed by two members of the program committee who will recommend a decision to the chair of the track that the paper has been submitted to. The track chair will make the final decision. Lastly, the Camera-Ready papers will be reviewed by one member of the program committee. Members of Program and Organizing Committees: The Program Committee includes members of chapters of World Academy of Science (chapters: supercomputing; scientific computing; artificial intelligence; imaging science; databases; simulation; software engineering; embedded systems; internet and web technologies; communications; computer security; and bioinformatics.) Many members of the program committee for individual conferences include renowned leaders, scholars, researchers, scientists and practitioners of the highest ranks; many are directors of large research laboratories, IEEE Fellows, heads/chairs of departments, deans and provosts. Each committee also includes two Student Members (in their final stages of their PhD programs) who are identified as such. Refer to the conference web sites for the list of members of program committee. Co-Sponsors (a partial list): Academic Co-Sponsors of WORLDCOMP'07 include: - Massachusetts Institute of Technology (MIT) Media Laboratory, MIT (Cambridge, Massachusetts) - Statistical Genomics and Computational Biology Laboratory, Department of Statistics, Harvard University (Cambridge, Massachusetts) - Texas Advanced Computing Center, The University of Texas at Austin (Austin, Texas) - Statistical and Computational Intelligence Laboratory of Purdue University (West Lafayette, Indiana) - University of Iowa's Medical Imaging HPC Lab (Iowa City, Iowa) - Institute for Informatics Problems of the Russian Academy of Sciences, Moscow, Russia); Other Co-sponsors include: - HPCwire - GRIDtoday - STEM Education Society - HPCSoft, HPC Software Inc. - International Technology Institute - H2cM - Hodges' Health, UK Purpose / History: CSC'07 will be held in conjunction with WORLDCOMP'07 (http://www.world-academy-of-science.org/worldcomp07/ws ). WORLDCOMP'07 is the largest gathering of researchers in computer science, computer engineering and applied computing. Many of the tracks of WORLDCOMP are considered to be the premier meetings for presentation of advances in their respective fields. We anticipate to have 2000 or more attendees from over 75 countries participating in the 2007 event. The motivation is to assemble a spectrum of affiliated research conferences into a coordinated research meeting held in a common place at a common time. The main goal is to provide a forum for exchange of ideas in a number of research areas that interact. The model used to form these annual conferences facilitates communication among researchers from all over the world in different fields of computer science, computer engineering and applied computing. Both inward research (core areas of computer science and engineering) and outward research (multi-disciplinary, inter-disciplinary, and applications) will be covered during the conferences. March 4, 2007: Submission of full/draft papers (about 5 to 8 pages) April 4, 2007: Notification of acceptance April 27, 2007: Camera-Ready papers and Registration due June 25-28, 2007: The 2007 International Conference on Scientific Computing (CSC'07) === Subject: Re: Mathematical Biology Joke a labour party economist. -- metro-golden-meower mhm x v i x i i i ,;S2GAAAA25r: .i#@@@@@@@@@@@@@@@@@@@#i, .r@@@@@@@@@@@@@@@@@@@@@@@@@@@@#s :3@@HXX&@@@@@@@@@@@@@@@@@@@@@@@@@@@@r :: .rH@@@@@@@@@@@@@@@@@@@@@@@@@3 ,9@@@@@@@@@@@H99@@@@@@@@@@@@@@@@@@@@@@@@@@@S ;@@@@@@@@@@@@@@@@@@#5::iH@@@@@@@@@i ,G@@: .@@@@@@@@@@@@@@@@@&; r@@@@@@@h .,sS r@@@@@@@@@@@@@@@@@#33#@@@@@@# ;@@@@@@@@@@@@@@@@@@@H i@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ s@@@@@@@@@@@@@@@@@@@S ;@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@X ,iB@@@@@@@@@@@@@A .@@@@@@@@@@@@@@@@@@@@@: ;5#A ,@@@@@@@@@@@@X @@@@@@@@@@@@@@@@@@@@@r r#@i :@2:@@@@@@@@@@@. r@@@@@@@@@@@@@@@@@@@@3 s@@@@@@& , @@@@@@@@@@@@ @@@@@@@@@@@@@@@@@@@@@, #@ i@@@sr@@r ;#@@@@@@@@@@: .@@@@@@@@@@r;@@@@@@@@@# ;@@, s. @@@@@@@@@# s@@@@@@@@r @@@@@@@@@@@@M;A@@ @r :;:. ;@@@@@@@@@@ A@@@@@@: .@@@@@@@@@@@@@@@@@ @@@i::siAG ,@@@@@@@ r@@@@@@@. @@@@@i .@@@@@@@@@@@@@@@@@ .@@@@@@@@@i S@@@. @@@@@@@r @@@@ @@@@@@@@@@@@@@@@@ @@@@@B: @ @@@@@@@2 M@@ &@@@@@@@@@@@@@@@@, @@@@. .h@@i @#s@@@@@@@@; 2@A ,@@@@@@@@@@2X@@@@X 2@@@ 5@@@@@@#G@@@@@@@@@@@@@ :@@B @@@@@@@@@@: @@@@@; : M@@ @@@@@@@@@@@@@@@@@@@@@@ @@@@ @@@@@@@@@ @@@@@@@ ,@ @@ 3@@@@@@@@@@@@@@@@@@@@@i 9@@@;:@@@@@@@@ s@@@@@@@@ :@s 3 @@@@@@#@@@@:M@@@@@@@@@ @@@,X@@@@@@@i B@@@@@@@@@@@@ 9@@@@@@ #@@@ :@@@@@@@@; ;@@ M@@@@@@@A H@@@@@@@@@@@@@. @@@@@@s @@@@2 @@@@@@@# #@ @@@@@@@@@3 ;@@@@@@@@@@@@@@@&: 2@@@@ @s ,S@@@@@@@ , #@@@@@@@@@@r@@@@@@@@@@@@@@@@@@@#i ;#@. ..iA9A@@@@@@ r@@@@@@@@@@@@@@@@@@@@@@@i#@@@@@@@@@G, ;&B@#r 9@@@@ @@@@@@@@@@@@@@@@@@@@@@@H h@@@@@@@@@@@# M@@@ @@@@@@@@@@@@@@@@@@@@@@@ H@@@@@@@@@@@@ i@@9 H@@@@@@@@@@@@@@@@@@@@@r @@@@@@@@@@@@; :@@@, .@@@@@@@@@@@@@@@@@@@@ H@@@@@@@@@@@ .M@@@r ;@@@@@@@@@@@@@@@@@r @@@@@@@@@@@r2@@@i 2@@@@@@@@@@@@@. 5@@@@@@@@@@,;@M. rH@@@@@@@A .@@@@@@@@@@r :5M#; :@@@@@@@MAr ;;. ARS GRATIA ARTIS *****************************PEDO ALERT**************************** (translation: 'everything without nappies, diapers to you dumb jank heads, can be stretched'.) (translation: 'sex is fun but it hurts. (lisa, 3 years old)'.) ***************************/PEDO ALERT***************************** meow