mm-2899 ==== Subject: Re: Prime challenge & fractional part > But the Mathworld plot you cite is { n log n } vs n, which seems > like a different kettle of fish. > Yeah, I haven't figured out why the patterns look so similar. > Still, it shouldn't be too hard > to work out what the patterns are that appear in the two plots, > and then see whether they are related in some way. > I'm definitely more interested in what the patterns themselves are > rather than determining if they are identical in the two plots. log (n + 1) = log n + log ( 1 + (1 / n) ) = log n + (1 / n) - (1 / (2 n^2)) + ... so (n + 1) log (n + 1) - n log n = log n + 1 + (1 / (2 n)) + .... so if log n is very close to an integer (i.e., when n is in the neighborhood of e^k for some positive integer k) the fractional part of n log n is changing very slowly - decreasing slowly while log n is less than the nearby integer, then increasing when log n passes the integer. When log n is close to a half-integer (k / 2 for odd k), {n log n} is changing by about .5 each time n increases. I think this explains a lot of what you're seeing, and no doubt a more careful analysis along these lines could get a more precise idea of the shapes. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) ==== Subject: Re: Counting binary words That is exactly what I was looking for. Good night, Rodolfo ==== Subject: Re: Slicing 4D matrix to 2D >Hello everyone, >I have a 4D Matrix A(x,y,p,q), where size(x)=size(y)=300, >size(p)=size(q)=30. Now I want to cut a 2D slice A(bx,by,cp,cq) - How >do I do that? I guess that your problem is inspired by arrays in computer programming languages. The question refers to classical elementary operations on multi-dimensional arrays in computer memory. To fix thoughts, let me use your example: A = array[{x}1:300, {y}1:300, {p}1:30, {q}1:30] of real; The compiler reserves a block of computer memory of size 300 * 300 * 30 * 30 * R bytes, where R is the byte length of a real number (commonly 8 or 10 bytes of 8 bits each). This memory block is provided with a so-called dope vector, a block of data containing... the starting address, the length of each array element, the lower and upper values of the indices, and the increments of storage addresses when one steps one unit in the... q index (increment = R, assuming that the last index counts array elements one by one, the next to last index counts rods of array elements, the 2nd next to last index counts slabs of array elements, etc.) p index: increment = 30 * R; y index: increment = 30 * 30 * R; x index: increment = 300 * 30 * 30 * R. With these data it is not too difficult to slice out... any array element A[x, y, p, q]; any rod of elements A[*, y, p, q] or A[x, *, p, q] etc., where * stands for the corresponding index running from lower limit to upper limit (or a subrange of the full range of that index); any slab A[*, *, p, q], in the x-y direction, or one of the other five main directions of planes in 4D space; any parallelepipedon A[*, *, *, q] or... etc., etc.. and finally the entire array A[*, *, *, *] or any 4D rectangular sub-array of A. I hope that this answers your question. Ciao: Johan E. Mebius ==== Subject: Re: obtaining notes on classic textbooks >>For me, the solutions are used to check my own solutions. > Suppose that you write a proof for some problem, and then you open the > solutions manual and find a completely different proof. How does that > help you tell whether your proof is correct? Just speaking from my own experience: Sometimes there will be enough similarities at least to show me whether I've been working in the right direction. Or, perhaps I'll spend some time working out *why* the book solution's approach was different from mine, and if perhaps in similar situations in the future I should use the book's approach instead. At the very least it may show me an alternate approach that might help with other proofs. But perhaps the most valuable thing to me is that it can keep me from getting permanently *stuck*. I'm the sort of person who has a very hard time making myself move on to question 5 if I haven't finished question 4 yet. I may spend hours -- or days -- trying to get past a particular problem without success. At such times it's really nice to be able to look at least a part of a solution and get some hints about how to proceed. That doesn't help me learn much if I do it every time I'm not sure about something, but after spending a *lot* of time wrestling with something it's nice to be able to look up a correct solution, figure out where I went wrong, and move on to something else. At least it avoids the situation of getting part way through a book, only to get irrevocably stuck, and end up putting the book away for a few days, or weeks, before coming back to it -- or possibly never finishing it at all. -- Wayne Brown (HPCC #1104) | When your tail's in a crack, you improvise fwbrown@bellsouth.net | if you're good enough. Otherwise you give | your pelt to the trapper. e^(i*pi) + 1 = 0 -- Euler | -- John Myers Myers, Silverlock ==== Subject: Re: obtaining notes on classic textbooks >> >For me, the solutions are used to check my own solutions. > >>Suppose that you write a proof for some problem, and then you open the >>solutions manual and find a completely different proof. How does that >>help you tell whether your proof is correct? >> >Just speaking from my own experience: Sometimes there will be enough >similarities at least to show me whether I've been working in the >right direction. Or, perhaps I'll spend some time working out *why* >the book solution's approach was different from mine, and if perhaps in >similar situations in the future I should use the book's approach instead. >At the very least it may show me an alternate approach that might help >with other proofs. >But perhaps the most valuable thing to me is that it can keep me from >getting permanently *stuck*. I'm the sort of person who has a very >hard time making myself move on to question 5 if I haven't finished >question 4 yet. I may spend hours -- or days -- trying to get past a >particular problem without success. At such times it's really nice to be >able to look at least a part of a solution and get some hints about how >to proceed. That doesn't help me learn much if I do it every time I'm >not sure about something, but after spending a *lot* of time wrestling >with something it's nice to be able to look up a correct solution, >figure out where I went wrong, and move on to something else. At least >it avoids the situation of getting part way through a book, only to get >irrevocably stuck, and end up putting the book away for a few days, or >weeks, before coming back to it -- or possibly never finishing it at all. A strong suggestion: Get over this do-it-in-orderness. Almost always, if you just move on, your brain keeps working in the background on the question you couldn't get. Then comes the aha moment as you brush your teeth. Besides, you can always ask for hints here on sci.math. As long as you assure people it is self-study and show what you've tried, people in the know are often very generous here. How did we ever survive before the Internet? -- Stephen J. Herschkorn sjherschko@netscape.net Math Tutor on the Internet and in Central New Jersey and Manhattan ==== Subject: Re: obtaining notes on classic textbooks > Typically, no such commentaries are available. The point of the > exercises is to spend the time to figure them out, until you reach > enlightenment. Once you have reached enlightenment, you will be able to > tell when your answers are correct. Until then, you have to take > classes so you can ask your prof if your proof seems right. But for those of us who aren't taking classes and depend entirely on the written material for *everything*, solution manuals are very helpful. It's frustrating to sweat for hours and hours and hours over a set of exercises, and then just have to *hope* the answers are correct, and wonder how much I really understand of what I *think* I know... -- Wayne Brown (HPCC #1104) | When your tail's in a crack, you improvise fwbrown@bellsouth.net | if you're good enough. Otherwise you give | your pelt to the trapper. e^(i*pi) + 1 = 0 -- Euler | -- John Myers Myers, Silverlock ==== Subject: How can the meaning of Goedel's unprovable statement descend from infinity? (Note: The author is aware that solutions of certain Diophantine equations are said to be mechanically uncheckable. This post focuses strictly on Goedel's Incompleteness Theorem, as it is proved by J.N. Crossley in What Is Mathematical Logic?) Don't you find it curious that the proof of Goedel's Incompleteness Theorem relies on free variables? When do these free variables ever appear in axioms or theorems in arithmetic? Why are they so important in proving the existence of a statement that cannot be proved or disproved? I have thought hard about Goedel's Incompleteness Theorem since I read a proof at age fifteen. My goal was to intuitively understand the meaning of the unprovable statement. I now believe that while the theorem is true, the statement is meaningless. It is just not the kind of statement we would wish to prove in arithmetic. In this post, I will attempt not a disproof of Goedel's theorem itself, but a proof of the meaninglessness of the unprovable statement. First, a little background information. The symbols of arithmetic (+, -, =, logic symbols, etc.) may be assigned certain numbers, so that statements in arithmetic may be coded into a Goedel number by making the assigned numbers exponents of the primes, in the order they occur. We can define what it means for a number to be the Goedel number of a proof: we make the Goedel numbers of sequentially derived statements exponents of the primes, in the order they are deduced. It is a proof if each deduction is valid, and this can be checked mechanically. Let x sub y be the Goedel number that results when all the free variables of the statement of the Goedel number x are replaced with the Goedel coding of the number y. In essence, x sub y refers to a function by which the value of y is filled in for the free variables in x. Let Pf(x, y) be *not* the property that x is the Goedel number of a proof of Goedel number y, but the *Goedel number of the statement* that x is the Goedel number of a proof of the Goedel number y. I give a brief outline of Crossley's proof here: Let g be the Goedel number of the statement ~(Ex)(Pf(x, y sub y)). This is read, There is no proof of y sub y. It contains one free variable, y, twice. Now consider the statement g sub g. One can easily see from the definition of g that it is precisely, There is no proof of g sub g. Thus, we have found a statement that is equivalent to the nonexistence of a proof of it. If it were provable, then it would be true, and a proof would not exist -- a contradiction. If it were disprovable, then there would be a proof -- another contradiction. It is therefore neither provable nor disprovable. Q.E.D. What I will attempt to show is that Goedel's unprovable statement is not a statement about meaningful propositions in arithmetic; rather, it is a statement about a meaningless statement about the substitution into itself of a Goedel number with two free variables. I will argue that it is meaningless precisely because it -- and all the equivalent statements that result when the repeated substitutions are carried out -- mention free variables in their nested statements. When the substitutions are carried out indefinitely, we arrive at a paradox of meaning I would call descent from infinity. For Goedel's unprovable statement to be a *meaningful statement in arithmetic*, there must be no free variables in any of the nested statements, and therefore the substitutions must be carried out indefinitely. But then we get a statement of the form: The following theorem is not provable: The following theorem is not provable: The following theorem is not provable: ... and so on, to infinity, which is clearly meaningless. First, I will make a few points. Goedel numbers can be about other Goedel numbers, which in turn may be about other Goedel numbers, and so on. A Goedel number with a free variable, such as that of x is prime, is meaningless because it is not a *statement about numbers*. It is only a statement about a possible number x. It is at best a property; it is not meaningful until we replace x with something. It should be remembered that there is not a single theorem in mathematics that explicitly contains a free variable. Although a statement can be about a meaningless Goedel number, it may still be meaningful. However, the statement it is *about* is meaningless. For example, the statement, the Goedel number of 'x is prime' is neither provable nor disprovable is a meaningful statement about a meaningless Goedel number. Now let me explain why Goedel's unprovable statement is meaningless. As in Crossley's proof, define g to be the Goedel number of the statement ~(Ex)(Pf(x, y sub y)). The unprovable statement is g sub g, and it is supposedly meaningful. What does it mean? Let us examine it more closely: g sub g = Goedel number of ~(Ex)(Pf(x, g sub g)). of clarity. Notice that g sub g is still about the statement g, and more importantly, it *still mentions a free variable* (hidden in g). What, then, does *this* statement mean? To decide its meaning, we must remove the free variable, and to do that, we must find the equivalent statement in which the substitution is already performed: g sub g = ~(Ex)(Pf(x, g sub g)) = ~(Ex)(Pf(x, ~(Ex)(Pf(x, g sub g)))) But this statement still mentions a free variable! And so on, to infinity. We are left with two choices. We must either admit that g sub g refers ultimately to a meaningless statement, or carry out the substitution process indefinitely, resulting in a paradoxical statement which is also meaningless. This proves the meaninglessness of Goedel's unprovable statement. All objections are welcome. No personal attacks. ==== Subject: Re: How can the meaning of Goedel's unprovable statement descend from infinity? > (Note: The author is aware that solutions of certain Diophantine > equations are said to be mechanically uncheckable. This post focuses > strictly on Goedel's Incompleteness Theorem, as it is proved by J.N. > Crossley in What Is Mathematical Logic?) The respondant has no clue what it might be that solutions of certain Diophantine equations are said to be mechanically uncheckable. Diophantine equations have solutions in whole numbers. Checking that values are, or are not, a solution of (even a system of) Diophantine equations requires only that the arithmetic operations indicated be performed. That's about as mechanical a calculation as I can think of. References, if you please? > Don't you find it curious that the proof of Goedel's Incompleteness > Theorem relies on free variables? When do these free > variables ever appear in axioms or theorems in arithmetic? Why are > they so important in proving the existence of a statement that cannot > be proved or disproved? Free variables are essential to the predicate calculus. Absent the axioms and axiom schema of the predicate calculus which allow free variables, I don't see how a formal logic can support bound variables. If you have in mind a nontrivial formal logic (certainly more than the propositional calculus) which dispenses with free variables, I interested to hear more about it. > I have thought hard about Goedel's Incompleteness Theorem since I > read a proof at age fifteen. My goal was to intuitively understand the > meaning of the unprovable statement. I now believe that while the > theorem is true, the statement is meaningless. It is just not the kind > of statement we would wish to prove in arithmetic. In this post, I will > attempt not a disproof of Goedel's theorem itself, but a proof of the > meaninglessness of the unprovable statement. You haven't been reading Gregory Chaitin's book, Meta Math: The Quest for Omega!, by any chance? Because your point here is quite similar to the autobiographical premise of that book. > First, a little background information. [snip preliminaries and an account of meaninglessness theorem] I'm not aware of any accepted formalization of meaninglessness. I'll be interested to take a closer look at your presentation, but it seems quite unlikely that a precise notion of meaningfulness or its absence is available. If you like historical background, you might look up the Fa-chia [Fa3 Jia1], the Legalist School of Chinese philosophy (circa 200AD) which spawned the ruthless First Emperor of China. Here Fa means law and chia is (roughly) philosophy. The connection is their principle that laws (and words) should have only one meaning. Seemingly a harmless principle, but in the course of events it led not just to book burning, but to the official persecution of scholars, many of whom were (according to legend) buried alive for the refusal to accept this doctrine. ==== Subject: Re: How can the meaning of Goedel's unprovable statement descend from infinity? > (Note: The author is aware that solutions of certain Diophantine > equations are said to be mechanically uncheckable. This post focuses > strictly on Goedel's Incompleteness Theorem, as it is proved by J.N. > Crossley in What Is Mathematical Logic?) > The respondant has no clue what it might be that solutions of > certain Diophantine equations are said to be mechanically > uncheckable. Diophantine equations have solutions in whole > numbers. Checking that values are, or are not, a solution of > (even a system of) Diophantine equations requires only that > the arithmetic operations indicated be performed. That's about > as mechanical a calculation as I can think of. References, if > you please? > Don't you find it curious that the proof of Goedel's Incompleteness > Theorem relies on free variables? When do these free > variables ever appear in axioms or theorems in arithmetic? Why are > they so important in proving the existence of a statement that cannot > be proved or disproved? > Free variables are essential to the predicate calculus. Absent the > axioms and axiom schema of the predicate calculus which allow > free variables, I don't see how a formal logic can support bound > variables. If you have in mind a nontrivial formal logic (certainly > more than the propositional calculus) which dispenses with free > variables, I interested to hear more about it. > I have thought hard about Goedel's Incompleteness Theorem since I > read a proof at age fifteen. My goal was to intuitively understand the > meaning of the unprovable statement. I now believe that while the > theorem is true, the statement is meaningless. It is just not the kind > of statement we would wish to prove in arithmetic. In this post, I will > attempt not a disproof of Goedel's theorem itself, but a proof of the > meaninglessness of the unprovable statement. > You haven't been reading Gregory Chaitin's book, Meta Math: The > Quest for Omega!, by any chance? Because your point here is > quite similar to the autobiographical premise of that book. > First, a little background information. > [snip preliminaries and an account of meaninglessness theorem] > I'm not aware of any accepted formalization of meaninglessness. > I'll be interested to take a closer look at your presentation, but it > seems quite unlikely that a precise notion of meaningfulness or > its absence is available. If you like historical background, you > might look up the Fa-chia [Fa3 Jia1], the Legalist School of > Chinese philosophy (circa 200AD) which spawned the ruthless > First Emperor of China. Here Fa means law and chia is > (roughly) philosophy. The connection is their principle that > laws (and words) should have only one meaning. Seemingly a > harmless principle, but in the course of events it led not just > to book burning, but to the official persecution of scholars, > many of whom were (according to legend) buried alive for the > refusal to accept this doctrine. Correction to my date on the Fa-chia and Qin Shi Huangi: It was ~200BC, not AD. More information here: http://en.wikipedia.org/wiki/Qin_Shi_Huang_Di -- chip ==== Subject: Re: How can the meaning of Goedel's unprovable statement descend from infinity? > I have thought hard about Goedel's Incompleteness Theorem since I > read a proof at age fifteen. My goal was to intuitively understand the > meaning of the unprovable statement. I now believe that while the > theorem is true, the statement is meaningless. It is just not the kind > of statement we would wish to prove in arithmetic. In this post, I will > attempt not a disproof of Goedel's theorem itself, but a proof of the > meaninglessness of the unprovable statement. > You haven't been reading Gregory Chaitin's book, Meta Math: The > Quest for Omega!, by any chance? Because your point here is > quite similar to the autobiographical premise of that book. I have not read that book, but the title sounds interesting. > First, a little background information. > [snip preliminaries and an account of meaninglessness theorem] > I'm not aware of any accepted formalization of meaninglessness. > I'll be interested to take a closer look at your presentation, but it > seems quite unlikely that a precise notion of meaningfulness or > its absence is available. I define a statement in arithmetic to be necessarily meaningless if it contains a free variable, in which case it is not a statement about numbers, but a mere property. ==== Subject: Re: How can the meaning of Goedel's unprovable statement descend from infinity? > I have thought hard about Goedel's Incompleteness Theorem since I > read a proof at age fifteen. My goal was to intuitively understand the > meaning of the unprovable statement. I now believe that while the > theorem is true, the statement is meaningless. It is just not the kind > of statement we would wish to prove in arithmetic. In this post, I will > attempt not a disproof of Goedel's theorem itself, but a proof of the > meaninglessness of the unprovable statement. > You haven't been reading Gregory Chaitin's book, Meta Math: The > Quest for Omega!, by any chance? Because your point here is > quite similar to the autobiographical premise of that book. > I have not read that book, but the title sounds interesting. > First, a little background information. > [snip preliminaries and an account of meaninglessness theorem] > I'm not aware of any accepted formalization of meaninglessness. > I'll be interested to take a closer look at your presentation, but it > seems quite unlikely that a precise notion of meaningfulness or > its absence is available. > I define a statement in arithmetic to be necessarily meaningless if it > contains a free variable, in which case it is not a statement about > numbers, but a mere property. I did spot your emphasis on excluding free variables from a logical system, but my comment about that was in the portion of my reply which you excised. I'm really not sure what common ground we can find to discuss your idea(s). Here's a stab: 1. In the predicate calculus, the use of variables is allowed. Each specific occurrence of a variable in a sentence (well- formed formula of a first-order formal language) is either free or bound. The latter is the case of a variable occurring in the scope of a quantifier, either existential (there exists x) or universal (for all x). Variables appearing not in the scope of a corresponding quantifier (for that variable) are said to be free occurrences. 2. If a sentence contains no free variables, it is a closed sentence. 3. Goedel's First Incompleteness Theorem (1931) gives a construction for a closed sentence S in the formal language of Peano Arithmetic (PA) about which it is shown that if PA is consistent, then S is not a theorem of PA, and if PA is omega-consistent, then ~S is not a theorem of PA. To put this in contrapositive terms, if S is a theorem of PA, then PA is inconsistent, and if ~S is a theorem of PA, then PA is omega-inconsistent. PA is omega-consistent means that if Q(x) is a sentence of PA with exactly one free variable x, and Q(n/x), the result of substituting a constant number n for x, is a theorem of PA for every constant number n, then there exists x s.t. ~Q(x) is _not_ a theorem of PA. 4. JB Rosser was later (1936) able to give a more complex construction which eliminates the can of worms opened by the possibility of omega-inconsistency. He showed the existence of a closed sentence T such that T is a theorem of PA if and only if ~T is a theorem of PA. In other words, either PA is inconsistent, or neither T nor ~T is a theorem of PA. 5. It follows from this combined result of Goedel & Rosser that PA is either inconsistent or incomplete, hence the given reference to the First Incompleteness Theorem. In order for this to be the case, it is essential that sentence T (and previously S) be closed sentences. Otherwise it would not constitute an incompleteness result. No one expects that a sentence with one or more free variables should either be true or false; in general the truth-value would depend on assignment of values (constants) to each free variable. So it is trivial to produce sentences which are not closed such that neither they nor their negation are theorems of PA, e.g. neither (x = 1) nor ~(x = 1) is a theorem of PA, unless PA is inconsistent. This is _not_ the result proved by Goedel & Rosser. 6. To cycle back to my earlier comment about free variables, I wish to know whether you are saying that no use of variables should be allowed, or whether you accept the use of variables, which are either free or bound as outlined above, but mistakenly jumped to a conclusion that Goedel and Rosser cheated by constructing a sentence that contained one or more free variables. If it is the latter, then I could better understand your concern, but indeed I would assure you that careful study will show their construction produces a closed sentence. ==== Subject: Re: How can the meaning of Goedel's unprovable statement descend from infinity? As I understand it, your argument is as follows: although Goedel did in fact prove that there exist statements which can be neither proven nor disporven, the statement he actually produced with this property is completely uninteresting. The reason this is so is that rather than being a complete logical sentence it contains free variables, and therefore does not express a property of numbers. You justify this by considering the formula g_g, where g is the formula g = ~(E x)(Pf(x, y_y)) so that when expanded, g_g means g_g = ~(E x)(Pf(x, g_g)) which requires further expansion to expunge the free variable g_g, and so on, so that the statement does not arrive at a precise meaning until it has been expanded infinitely, at which point you claim the resulting statement is paradoxical. I first of all disagree with you that the statement is meaningless, and I think the reason your understanding is flawed is that you do not quite understand the real necessity of the Goedel numbering system to the proof. This may itself have something to do with the fact that you are misusing the phrase free variable: in logic, a variable is a symbol, and a free variable is a symbol which is not quantified within its scope. The sense you are using it seems to be that g_g is a free variable in the above expression because it appears in its own literal expansion, rather than being defined purely by basic logical entities. This apparent self-reference is exactly the reason Goedel numbers are necessary to make the proof work, and when they are used correctly, the definition of g_g is not in fact self-referential: you don't need to substitute g_g into itself to get an arithmetic statement. Here's how I understand the proof. The end goal is to be able to write a sentence which states This sentence cannot be proven. The problem is that in formal logic, if one wants to talk about a specific object one needs to be able to define it; that is, produce a formula P(x) with one free variable (this is free variable in the sense I explained it, not the sense you are using it) which is satisfied only by the object you desire to discuss. In order to apply a logical formula to another logical formula, you need to interpret such formulae as objects of your logical system. Once you do that, you still have to be able to state in the language of formal logic what it means that a statement is provable. That is, is there some kind of a relation like equality, divisibility, etc., which determines whether a statement can be proven? This relation has to be defined using only the axioms of your logical system, in this case arithmetic. The first part of the program is to make logical formulae into something that arithmetic can discuss; namely, numbers. That's what the Goedel numbering is: every formula is a number, so we can talk about formulae by talking about their associated numbers. In fact, and this is the key point, if you forget momentarily the fact that the Goedel numbers correspond to logical statements, then any statements you make about these numbers are simply expressions in number theory. With an eye towards the proof, this means that when it comes time to write a sentence about itself, we can get away with writing a sentence about a number, and that number will just happen to be the Goedel number of the sentence it's in. Another important fact about Goedel numbers, which is where all the discussion of prime factorizations comes in, is that the transformation that takes a formula and produces its Goedel number is itself logically definable. If you like, you may think of formulae as strings of numbers (replace each symbol by its number), so that the Goedel-numbering function is just a function on numbers written in, say, decimal notation. That is, it's an arithmetic function. This means that the Goedel number construction used in the proof is actually an arithmetic function, rather than an operation on logical statements (although it has _implications_ for logic). Also, in what follows I _will_ write all the occurrences of Goedel number of... which you omitted, since it clarifies what's a number and what's a formula. Thus, n(x) means The Goedel number of x, which is an arithmetically defined function. Note that the sub operation x_y is subtle, and occurs in two contexts: numerical context and logical context. Although 'x' and 'y' are _intended_ to be formulae, they _are_ simply variables. The sub operation as applied to two numbers m and p performs the arithmetic operation on them that would be performed on two Goedel numbers n(x) and n(y), namely the operation which corresponds to the formal substitution of n(y) into the free variables of a formula 'x' and then taking the Goedel number of the result. The point here is that the formal substitution of a number into a free variable of a formula can itself be formalized as an arithmetic function. However, in the eventual proof, sub is used not only in a formula but as an operation on formulae: we produce g_g from g by substituting n(g) into the free variable of g. In this case, 'g' is an honest logical formula and our ability to do this operation is simply an axiom of logic that says we can substsitute any constant expression into any free variable. The second part of the program is to talk about provability of logical statements as a property of their Goedel numbers. After various contortions one arrives at the conclusion that the relationship Pf between two numbers (in particular, the Goedel number of a proof and the Goedel number of a formula) is in fact definable in arithmetic; that is, there's a logical formula which for any string of formulae x and any formula y, when you plug in n(x) and n(y) tells whether x is a proof of y; the important fact is that Pf takes numbers as arguments, not formulae. Therefore it's a logical object. Now the sentence g = ~(E x) ( Pf(x, y_y) ), which has one free variable, can be written down. That's the hard part. What we accomplish by doing this is to obtain a formula which, if we insert any Goedel number into the free variable 'y', tells us (by the truth or falsity of the resulting sentence) whether or not the sentence of which 'y' is the Goedel number, has a proof. Now I repeat; g is a _formula_ of logic, merely a string of symbols. It has a Goedel number, and that's an integer. When we form g_g we replace y by g in the above expression, which means first we compute its Goedel number, then we substitute that number into g in all the places that y occurs as a free variable (which are hiding inside the very complicated arithmetic operation y_y), at the end of which we have no free variables left, just a lot of symbols and numbers; this gives us g_g. Now, g_g is also a sentence, since in arithmetic, all numbers are logical objects. And as you noted, g_g states that it does not have a proof, and also implies that it does not have a disproof. _This is not circular_. There are no unresolved variables left in g_g; the only free variable in g was 'y', and it got filled in with some number when we made g_g. The fact that this number is actually the Goedel number of g itself is not relevant if we are simply wondering whether g_g is a sensible statement; it is, because it consists only of logical symbols and numbers. It is the intricate web of relations that tie together the Goedel-numbering scheme, the Pf function, and the actual logical meaning of a sentence, which ends up leading to g_g talking about its own unprovability. That's as much as I can say on the subject. Goedel numbering is like two people talking behind each others' backs: person A says Person B never lies and person B says Person A is an idiot, and the net result is that Person A insults himself, without ever knowing that's what he's doing. Actually, though, Goedel's proof is more airtight than this, since in addition it has a way of talking about what it means for a person to lie without actually having to talk about people. Well, there is one more thing I can say. You are right in that the to identify in practice. In later years people have identified actual statements which are independent of a logical system (like the continuum hypothesis). -- Ryan Reich ryan.reich@gmail.com ==== Subject: Re: How can the meaning of Goedel's unprovable statement descend from infinity? However, despite your disagreement and having accused my understanding of being flawed, you have not offered a valid objection to my proof of the meaninglessness of Goedel's unprovable statement, as I have defined it. My argument is simple. g sub g (or, as you call it, g_g) is not the free variable. g_g can be expanded as ~(Ex)(Pf(x, y_y)) sub ~(Ex)(Pf(x, y_y)). After substitution, we have ~(Ex)(Pf(x, ~(Ex')(Pf(x', y_y)) sub ~(Ex')(Pf(x', y_y)))). Here, ~(Ex')(Pf(x', y_y)) sub ~(Ex')(Pf(x', y_y)) is Goedel's unprovable statement. It *clearly* contains a free variable. Because statements with free variables are meaningless, this statement is meaningless. It is not interesting that there is no proof or disproof, any more than it is that there is no proof or disproof of the statement x is prime. It is not even interesting when one makes an additional substitution, which gives us ~(Ex)(Pf(x, ~(Ex')(Pf(x', ~(Ex'')(Pf(x'', y_y)) sub ~(Ex'')(Pf(x'', y_y)))))). This statement is read, There is no proof that there is no proof of ~(Ex'')(Pf(x'', y_y)) sub ~(Ex'')(Pf(x'', y_y)). Because its core statement, ~(Ex'')(Pf(x'', y_y)) sub ~(Ex'')(Pf(x'', y_y)), *contains a free variable*, it is only a meaningful statement about a meaningful statement about a *meaningless* statement. Statements about the nonexistence of proofs or disproofs of meaningless statements are not interesting. I will comment on your post below. > fact prove that there exist statements which can be neither proven nor > disporven, the statement he actually produced with this property is > completely uninteresting. The reason this is so is that rather than > being a complete logical sentence it contains free variables, and > therefore does not express a property of numbers. You justify this by > considering the formula g_g, where g is the formula > g = ~(E x)(Pf(x, y_y)) > so that when expanded, g_g means > g_g = ~(E x)(Pf(x, g_g)) > which requires further expansion to expunge the free variable g_g, > and so on, so that the statement does not arrive at a precise meaning > until it has been expanded infinitely, at which point you claim the > resulting statement is paradoxical. It is more correct to say that the free variable is *in* g_g. Other than that, you have understood me completely up to now. > I first of all disagree with you that the statement is meaningless, and > I think the reason your understanding is flawed is that you do not > quite understand the real necessity of the Goedel numbering system to > the proof. This may itself have something to do with the fact that you > are misusing the phrase free variable: in logic, a variable is a > symbol, and a free variable is a symbol which is not quantified within > its scope. The sense you are using it seems to be that g_g is a free > variable in the above expression because it appears in its own literal > expansion, rather than being defined purely by basic logical entities. > This apparent self-reference is exactly the reason Goedel numbers are > necessary to make the proof work, and when they are used correctly, the > definition of g_g is not in fact self-referential: you don't need to > substitute g_g into itself to get an arithmetic statement. It is not my argument that g_g is self-referential. I claim that g_g contains a free variable and is therefore meaningless. [beginning of proof snipped] > Now the sentence > g = ~(E x) ( Pf(x, y_y) ), > which has one free variable, can be written down. That's the hard > part. What we accomplish by doing this is to obtain a formula which, > if we insert any Goedel number into the free variable 'y', tells us (by > the truth or falsity of the resulting sentence) whether or not the > sentence of which 'y' is the Goedel number, has a proof. Now I repeat; > g is a _formula_ of logic, merely a string of symbols. It has a Goedel > number, and that's an integer. When we form g_g we replace y by g in > the above expression, which means first we compute its Goedel number, > then we substitute that number into g in all the places that y occurs > as a free variable (which are hiding inside the very complicated > arithmetic operation y_y), at the end of which we have no free > variables left, just a lot of symbols and numbers; this gives us g_g. > Now, g_g is also a sentence, since in arithmetic, all numbers are > logical objects. And as you noted, g_g states that it does not have a > proof, and also implies that it does not have a disproof. > _This is not circular_. There are no unresolved variables left in g_g; > the only free variable in g was 'y', and it got filled in with some > number when we made g_g. You misunderstood my argument about the problematic nature of statements that *mention* free variables. A statement can *mention* a free variable without *containing* one. As I said before, g_g can be expanded as ~(Ex)(Pf(x, y_y)) sub ~(Ex)(Pf(x, y_y)). This statement does not contain a free variable, but it is *about* a statement that *does*. The free variable is in the y_y of the second statement that replaces. > The fact that this number is actually the > Goedel number of g itself Please explain how g_g is the Goedel number of g. g_g is clearly a *longer string of symbols* that *contains g*, so it must be larger. The same can be said about g_g in relation to ~(Ex)(Pf(x, g_g)). The latter is clearly a longer statement containing the former, and thus it has a larger Goedel number. Statements can be equivalent while having different Goedel numbers. > is not relevant if we are simply wondering > whether g_g is a sensible statement; it is, because it consists only of > logical symbols and numbers. It is the intricate web of relations that > tie together the Goedel-numbering scheme, the Pf function, and the > actual logical meaning of a sentence, which ends up leading to g_g > talking about its own unprovability. > That's as much as I can say on the subject. Goedel numbering is like > two people talking behind each others' backs: person A says Person B > never lies and person B says Person A is an idiot, and the net > result is that Person A insults himself, without ever knowing that's > what he's doing. Actually, though, Goedel's proof is more airtight > than this, since in addition it has a way of talking about what it > means for a person to lie without actually having to talk about people. > Well, there is one more thing I can say. You are right in that the > to identify in practice. In later years people have identified actual > statements which are independent of a logical system (like the > continuum hypothesis). > -- > Ryan Reich > ryan.reich@gmail.com ==== Subject: Re: How can the meaning of Goedel's unprovable statement descend from infinity? > However, despite your disagreement and having accused my understanding > of being flawed, you have not offered a valid objection to my proof > of the meaninglessness of Goedel's unprovable statement, as I have > defined it. > My argument is simple. g sub g (or, as you call it, g_g) is not the > free variable. > g_g can be expanded as > ~(Ex)(Pf(x, y_y)) sub ~(Ex)(Pf(x, y_y)). Recall what this means: to take the second formula, find its Goedel number, and replace all occurrences of y in the first formula with n. Take note of my paragraph on the subtelties of the substitution function, since it's very relevant here. There are actually two uses back-to-back and they are confusing. The first use is the lexical substitution: you take the Goedel number of the second formula and physically insert it in place of all occurrences of y in the first formula. The second is numerical substitution: that's what is happening in the y_y there; after you have lexically substituted, this is just g(n) sub g(n) a function of numbers and not formulae. You don't get y back in this expression; you just get a number. Let me give a short, artificial example of what's going on here. Let p = (E x) (x = y) which is obvious and also not of interest in the same way as the Goedel sentence, but it does contain a free variable. It's got a Goedel number, say N, and using this we can form p sub p: p sub p = (E x) (x = N). This does not contain a free variable. It does not even make sense to look for one by expanding N to get back p, like (E x) (x = (E x) (x = y)) since it is meaningless for a variable to equal a sentence; this is no longer a well-formed formula (this is discussed in any introductory text on formal logic). > After substitution, we have > ~(Ex)(Pf(x, ~(Ex')(Pf(x', y_y)) sub ~(Ex')(Pf(x', y_y)))). > Here, ~(Ex')(Pf(x', y_y)) sub ~(Ex')(Pf(x', y_y)) is Goedel's > unprovable statement. It *clearly* contains a free variable. I'd like to note here that this statement does *not* clearly contain a free variable. By definition of sub, this expression actually means that you take the Goedel number of the latter formula and substitute it into all occurrences of the free variable in the former formula. The free variables are eliminated. Now, the way you've written the expression it looks like it has a free variable, but you haven't written it in a logically formally correct way. When you do, the free variable vanishes. It's clear that you didn't really understand what I was saying. It may well be that the original formula, g = ~(E x)( Pf(x, y_y) ) contains the free variable y. However, its Goedel number does not. It does not even make reference to y. It is simply a number. Hence, when n(g) is plugged into y in g, the free variable disappears. It may seem that g_g contains a free variable when you unravel the Goedel numbers in it, but that is a misuse of formal logic. g_g is simply a statement about numbers. When you see a statement like that you try to understand it by reducing it to simpler statements and understanding their relationships. In doing this, you naturally are led to unravel the Goedel numbering, back-substitute the corresponding formulae, and obtain what appears to be an unresolved free variable. However, this is wrong in two ways: first, in the way that I indicated above, that a Goedel number is not a formula but a code for a formula, and that even if the formula it codes has a free variable the Goedel number itself does not, being just a number. The second way it's wrong is much the same way that people who claim that 0.999... and 1 are not equal are wrong. They often argue that there is a slight difference, that 0.999... is just a little less all the way at the end. The thing is, you never get to the end. And when you try to unravel the Goedel sentence you are trying to achieve the same thing. You never actually see the free variable; it's hidden in another expression, and in fact there's no way to write any stage of the iteration so that y explicitly appears. At each stage, it's just a reference inside another Goedel number that you must unravel to look any farther. If you can't find it at any finite stage, however, it is not there. Formulas cannot be infinite. > _This is not circular_. There are no unresolved variables left in g_g; > the only free variable in g was 'y', and it got filled in with some > number when we made g_g. > You misunderstood my argument about the problematic nature of > statements that *mention* free variables. A statement can *mention* a > free variable without *containing* one. As I said before, g_g can be > expanded as > ~(Ex)(Pf(x, y_y)) sub ~(Ex)(Pf(x, y_y)). > This statement does not contain a free variable, but it is *about* a > statement that *does*. The free variable is in the y_y of the second > statement that replaces. This is different than what you seemed to be saying above. Would you say that the sentence (E x) (x = 2) is meaningless because if you write it like (E x) (x = y) sub 2 (where for simplicity I've chosen the Goedel numbering so that 2 is its own Goedel number) it is about a statement with a free variable? What about (E x) (x = y) sub n(y) where n(y) is whatever the Goedel number of the symbol y is. Actually, let's say y has Goedel number 5. Then this sentence is (E x) (x = 5) which is again true, and meaningful, although it expands to an expression that concerns free variables. Tell me how you can distinguish between this and the situation with the Goedel sentence. > The fact that this number is actually the > Goedel number of g itself > Please explain how g_g is the Goedel number of g. g_g is clearly a > *longer string of symbols* that *contains g*, so it must be larger. I think you misunderstood me here. The Goedel number of g is what you substituted for y within g; the resulting sentence is g_g. > The same can be said about g_g in relation to ~(Ex)(Pf(x, g_g)). The > latter is clearly a longer statement containing the former, and thus it > has a larger Goedel number. > Statements can be equivalent while having different Goedel numbers. However, I made no attempt to argue based on a relationship between Goedel numbers and truth values of statements. My point was simply that once you have replaced y with the Goedel number of g, the fact that the expression you get has a number in it which codes a sentence with a free variable is irrelevant; there is not actually a free variable *in* the resulting expression. If you want to argue that to assign meaning to a sentence you must unravel any codes in the sentence and see if they are secretly meaningless, you could easily conclude that no sentence has meaning, since it's trivial to invent a code that turns any meaningless statement you like into any expression you want. The point of formal logic is to make manipulations that do *not* take internal structure into account, but merely proceed by manipulation of formal expressions. In this sense, the brilliance of Goedel's proof is that it shows that despite this fact, you can perform this manipulation in the context of arithmetic, which hides the metalanguage of formal logic behind still more formal manipulations. Once you can put the metalanguage, the language of talking about proofs, into the machinery of doing proofs within the formal language itself, you can write a formal expression whose very existence takes the indirect route through arithmetic to have meta-meaning. -- Ryan Reich ryan.reich@gmail.com ==== Subject: Re: How can the meaning of Goedel's unprovable statement descend from infinity? > I have thought hard about Goedel's Incompleteness Theorem since I > read a proof at age fifteen. My goal was to intuitively understand the > meaning of the unprovable statement. I now believe that while the > theorem is true, the statement is meaningless. It is just not the kind > of statement we would wish to prove in arithmetic. Interesting. Godel showed, presumably, that if a formal system is consistent, then there is no proof within that formal system that proves that the formal system is consistent. However, if we insist that the notion of proof must capture an intuitive notion of a compelling argument, then we realize that if we believe that the formal system can prove anything at all, then at a minimum, we must already believe that the formal system is consistent, (otherwise it could formally prove anything, and hence it could prove nothing in the sense of a compelling argument) -- the point being that Godel's theorem is already implicit in our notion of belief. So maybe Godel's theorem is nothing more than circular nonsense. However, currently, it is regarded as one of the biggest advances in the field of logic in the twentieth century, and any amateur who questions that will be denounced as a crackpot. So my advice would be, unless you are certain that you want to get involved with this topic, just ignore it and stay sane. know if there is any argument of value there. ==== Subject: linear operator, determinant Let V be the vector space of all nxn matrices over the field of complex numbers, and let B be a fixed nxn matrix over C. Define a linear operator M_B on V by M_B(A) = BAB*. Show that det M_B = |det B|^{2n}. The determinant of M_B is equal to that of the matrix P representing M_B in an ordered base S for V. Let E_{xy} be a matrix in V such that [E_{xy}]_{ij} = 1 if (i,j)=(x,y), 0 otherwise. Let S = {E_{11}, E_{12}, ..., E_{1n}, E_{21}, ...., E_{2n}, .........., E_{n1}, ..., E_{nn}} be an ordered base of V. First of all I start to determine the generic element ij of BE_{xy}B*: [BE_{xy}B*]_{ij} = Sum_k [B E_{xy}]_{ik}{B*}_{kj} = = Sum_k ~B_{jk}[B E_{xy}]_{ik} = = Sum_k ~B_{jk}Sum_h B_{ih}[E_{xy}]_{hk} = = ~B_{jy}Sum_h B_{ih}[E_{xy}]_{hy} = = ~B_{jy}B_{ix}[E_{xy}]_{xy} = = ~B_{jy}B_{ix} Now I am in big trouble. How can I write P? I could write P_{st} = ~B_{jy}B_{ix} where s = (i-1)n + j and t = (x-1)n + y but what should I do now? Should I write P explicitly hoping for a miracle or should I use the general determinant formula (the sum extended over the permutation) expecting some simplification? Kiuhnm ==== Subject: Re: linear operator, determinant Kiuhnm ha scritto: > Should I write P explicitly hoping for a miracle or should I use the > general determinant formula (the sum extended over the permutation) > expecting some simplification? Since |detB|^{2n} = (detB~detB)^n = det(B~B)^n, I would have expected P = B~B 0 0 0......0 0 B~B 0 0......0 0 0 B~B 0......0 . . . ........ . . . .......0 0 0 0 0......B~B The previous exercise was simpler (M_B(A) = BA) and I found P = B B B B . . B then detP = (detB)^n. Kiuhnm ==== Subject: Re: linear operator, determinant I haven't worked out the details, but I seem to recall that the determinant is the product of the eigenvalues. I wonder if you can write the eigenvalues for M_B in terms of the ones for B. ==== Subject: Re: linear operator, determinant Sean Holman ha scritto: > I haven't worked out the details, but I seem to recall that the determinant is the product of the eigenvalues. I wonder if you can write the eigenvalues for M_B in terms of the ones for B. I cannot use eigenvalues because I do not know them and the authors of the exercise, Hoffman & Kunze, are well aware of that. Kiuhnm ==== Subject: Elemental study of the last theorem of Fermat More than 80 properties in: http://www.telecable.es/personales/carloman/ ==== Subject: Elemental study of the last theorem of Fermat IN http://www.telecable.es/personales/carloman/ ==== Subject: Re: Actual surrogate factoring solution equations [JSH] ... >> But you see, I know the equations are correct, I know they are not >> generally solvable in terms of factors of T, and I can't find a >> mathematical reason why they do not work. [David C. Ullrich] > What?????????????? > Lemme see if I got this straight. You have this amazing factorization > method. It's going to make you famous and destroy the world economy at > that same time. But it _doesn't work_, and you can't figure out why > not? That's not the story yet. While it in fact doesn't work worth beans (it's _possible_ to stumble into an integer factorization using it, but seems extremely unlikely that stumble can be improved), James doesn't know that yet. He _could_ know it simply by trying to use it, but he's too lazy (or too concerned about future generations, or too worried about saving his own ass, or something) to take that step. Instead, he knows it must work (so why bother trying it?). The reason he knows is the reason he gave: he can't find a mathematical reason for why it doesn't work. Therefore it must work. Insert JSH's flavor of QED. All of these surrogate factoring thingies have a lot in common. James pulls some system of equations out of his inexhaustible bag of advanced mathematical knowledge, and arrives at a formal factorization: T = f*g where T is the integer to be factored, and f and g are more-or-less God-awful algebraic expressions, usually with an absurd number of strangely named unknowns. Some subexpression of f and/or g gets called the surrogate, and you proceed by plugging in all ways of factoring the surrogate, hoping that's enough for the full f and/or g to reveal an integer factor of T. The problem in real life is that the expression for f doesn't necessarily resolve to an integer. Or to a rational (although he had some methods last year where it did). This time around it's not even limited to the reals! In the one example James tried to work out this time, starting with T=119 he managed to slog through his equations to derive that 100*T was the product of two specific complex numbers. That result is useless for finding that 119=7*17, but James doesn't see that as being a problem. Instead he argues that the equations doen't care whether they produce rational or irrational factors, and therefore have an equal chance of producing either. It's impossible to explain to him why that line of argument is absurd (several tried, in extreme detail, last year), so he keeps acting as if it were true. Therefore he makes no progress, just digging himself deeper into the same old hole. He can understand integer counterexamples provided the integers are small enough, so that's apparently still the only thing that can shake him loose. Alas, looks like he found that trying to factor 119 with his method was so tedious that he gave up before discovering that his current method _usually_ produces useless factorizations, and more often the larger T gets (a feature common to all surrogate factoring methods to date). > Curiouser and curiuouser. Most people wouldn't regard a method that > _doesn't work_ as an amazing breakthrough. We didn't get there with this one yet. That _was_ the fate of his generalized surrogate factoring theorem from last year, though (a convoluted way of generating rational factors of an integer, which was said (loudly ;-) to be a breakthrough despite that every non-zero rational is a rational factor of every integer). ==== Subject: Re: Actual surrogate factoring solution equations <11v543phjibta9e@corp.supernews.com> I say; soon, we will all dig ourselves out on the other side -- g'day, mates! thus quoth: were true. Therefore he makes no progress, just digging himself deeper into the same old hole. He can understand integer counterexamples provided the thus: eh, he's probably just a victim of algebraware, which might make him a Wolframite.... incidentally, you may have been quoting Hobbs, his no royal road to geometry, who didn't know about Bucky Fuller, then. (er, see the last URL in my sig .-) thus: Moving Dimensions Theory is *brand* new; eh?... you should read Smolin's book for a nice, low-key dyscussion of the quality of stringtheory; he's a looptheorist, though.... wow; maybe some one'd take you seriously, then. --Give Earth a Trickier Dick Cheeny -- out of office, after gigayears! http://tarpley.net/bush8.htm http://www.benfranklinbooks.com/ http://members.tripod.com/~american_almanac http://www.wlym.com/pdf/iclc/howthenation.pdf http://www.rand.org/publications/randreview/issues/rr.12.00/ http://www.rwgrayprojects.com/synergetics/plates/figs/plate01.html ==== Subject: Re: contour integrals > Ohh wait. The smaller one gives the required value because in the > limit, the expression being integrated approches 1/z, so its integral > is ln z. The other approaches 0 because the thing being integrated > there approches 0. Ha! I should have seen that earlier. Well, still no. You need to understand what a contour integral is. The integral over the large semicircle C, int_C e^(iz)/z dz, is int_[0,Pi] [e^(iRe^(it))/(Re^(it))]*iRe^(it)dt. You get that by setting z = Re^(it), 0 <= t <= Pi. That simplifies to i*int_[0,Pi] e^(iRe^(it)) dt, but there's still work to do. ==== Subject: Minimum of f(x) I'm trying to find out the minium of the function f(x) = sin(tan(x)-tan(sin(x)) where the interval x is betwen 0 and pi. I was told it's difficult to compute but I was able to do it in matlab and I'm trying to figure out mathematically why it would be considered hard to plot. Would the value also change if we allow the limit of x -> pi/2 ==== Subject: Re: Minimum of f(x) > I'm trying to find out the minium of the function f(x) = > sin(tan(x)-tan(sin(x)) where the interval x is betwen 0 and pi. I was > told it's difficult to compute but I was able to do it in matlab and > I'm trying to figure out mathematically why it would be considered hard > to plot. Would the value also change if we allow the limit of x -> pi/2 tan x blows up at pi/2 and 3 pi/2 How do you handle that case ? ==== Subject: Re: Minimum of f(x) > I'm trying to find out the minium of the function f(x) = > sin(tan(x)-tan(sin(x)) where the interval x is betwen 0 and pi. I was > told it's difficult to compute but I was able to do it in matlab and > I'm trying to figure out mathematically why it would be considered hard > to plot. It's not hard to plot. But plotting is different from computing. With a plot the best you can do is estimate. Computing means you are able, given sufficient computation time, to get the value to any desired precision. Do you know what the minimum is to, say, 6 decimal digits of precision? Probably what you were told is that it is difficult to compute analytically (by taking the derivative, setting it equal to 0, and solving the resulting equation). It wouldn't be difficult to calculate with an iterative numerical procedure. - Randy ==== Subject: Re: Minimum of f(x) > I'm trying to find out the minium of the function f(x) = > sin(tan(x)-tan(sin(x)) where the interval x is betwen 0 and pi. I was > told it's difficult to compute but I was able to do it in matlab and > I'm trying to figure out mathematically why it would be considered hard > to plot. > It's not hard to plot. But plotting is different from computing. > With a plot the best you can do is estimate. Computing means > you are able, given sufficient computation time, to get the > value to any desired precision. > Do you know what the minimum is to, say, 6 decimal digits > of precision? > Probably what you were told is that it is difficult to compute > analytically (by taking the derivative, setting it equal to 0, and > solving the resulting equation). It wouldn't be difficult to calculate > with an iterative numerical procedure. I'd like to see it done that way, Randy! The function f(x) = sin(tan(x)) - tan(sin(x)) has infinitely many extrema on the interval [0, pi] and does not achieve an absolute minimum there. Getting closer and closer to x = pi/2, one can, however, find relative minima which are closer and closer (but never equal) to -1 - tan(1), roughly -2.5574077. David ==== Subject: Re: Well Ordering the Reals Virgil said: >> That means that there are a whole bunch of strings of uncountably many >> digits that are not TO-numbers at all. What are all those non-TO-number >> strings doing there? > What are all the irrationals doing among the reals? Can you digitally represent > any of them exactly? Many of them, in fact. For example: x = 0.1234567891011121314151617181920... x is irrational (named after some mathematician, in fact), and is exactly representable by a nonrepeating infinite sequence of decimal digits. Predictable, too, because we can determine exactly what the nth digit is for any n. Here's another, in binary: y = 0.1101001000100001000001000001... Same thing for pi; there is an equation for determining the nth base-16 fractional digit of pi given any integer n. ==== Subject: Re: Well Ordering the Reals David R Tribble said: >> That could be your first useful proof. Take an arithmetic problem that >> is difficult or impossible using p-adics and show how easier it is >> using T-riffic numbers. > Well, we could talk about how the 10-adics ...111, ...112, and ...113 are all > apparently evenly divisible by 7, which makes no sense, whereas 1:000...000 > when divided by 7 has 6 possible remainders, none of them 0, which makes > perfect sense. You're not remembering correctly. Someone proved that several of your numbers, 0:111...110+k or some such, was divisible by 7 because a) they have a leftmost digit, and b) the number of digits is indeterminate. The p-adics don't have this problem, because the arithmetic is formally well-defined for them. ==== Subject: Re: Sting Theory's Postulates, Laws, and Experimentally Verified Equations. Waht? There Are None? Then Why Do We Support It With Billions? http://www.nsf.gov/awardsearch/piSearch.do;jsessionid=12140160808182889516AF 72BFC8F425?SearchType=piSearch&page=1&QueryText=string&PIFirstName=&PILastNa m e=&PIInstitution=&PIState=&PIZip=&PICountry=&Search=Search#results Billions ==== Subject: Re: Sting Theory's Postulates, Laws, and Experimentally Verified Equations. Waht? There Are None? Then Why Do We Support It With [sic] Billions? > http://www.nsf.gov/awardsearch/piSearch.do;jsessionid=12140160808182889516AF 7 2BFC8F425?SearchType=piSearch&page=1&QueryText=string&PIFirstName=&PILastNam e =&PIInstitution=&PIState=&PIZip=&PICountry=&Search=Search#results > Billions The average grant is somewhere around $200,000; maybe about 600 of the 900+ listings pertain to research in string theory (e.g. some are for research involving string as in the computer science term 'string) ... that adds up to, maybe 120 million dollars. The listings are over 25 years, which averages maybe 5 million dollars a year. You've got a point: the NSF should be funding more research in science and math education -- particularly that on: how do actually do math and on teaching the general population the difference between billion and million, so that they don't go running off screaming like headless chickens every time a perceived indignity ends up happening that resides nowhere except in their own imaginary, math-illiterate notions of the world. ==== Subject: Re: Sting Theory's Postulates, Laws, and Experimentally Verified Equations. Waht? There Are None? Then Why Do We Support It With [sic] Billions? Are you kidding me? Take that $120,000. Add the billions more of tax dollars that go to support the string theorists' offices. The health befenefits for their husbands, wives, and children. Their summers off in Russia. Their visits to ski chalets. And the total quickly comes to billions. String theory has 33 dimensions. So do the math. That's $30,000,000 per dimension. Even the curled up ones are expensive. All I'm saying is that more of that money should be gong to grad students and postdocs who are doing real science. ==== Subject: Re: Sting Theory's Postulates, Laws, and Experimentally Verified Equations. Waht? There Are None? Then Why Do We Support It With [sic] Billions? Sting Theory's Postulates, Laws, and Experimentally Verified Equations. Waht? There Are None? Then Why Do We Support It With Billions? ************* Sting has a theory? What is it? That rock stars make an unconscionable amount of money, but I'm getting mine? ==== Subject: Re: Sting Theory's Postulates [meaning the health care crisis], Laws, and Experimentally Verified Equations. Waht? There Are None? Then Why Do We Support It With [sic] Billions? > Are you kidding me? > Take that $120,000. > Add the billions more of tax dollars that go to support the string > theorists' offices. > The health befenefits for their husbands, wives, and children. So the other shoe drops. This has nothing to do with science funding at all, but the rising cost of health care. > All I'm saying is that more of that money should be gong to grad > students and postdocs who are doing [sic] real science. And who are you to say which is real and which is not? Most of those funds probably ARE going to grad students, postdocs, or other teams they are affiliated with. ==== Subject: Re: Sting Theory's Postulates [meaning the health care crisis], Laws, and Experimentally Verified Equations. Waht? There Are None? Then Why Do We Support It With [sic] Billions? String theory is real. Any other theory is not. MDT MDT MDT ==== Subject: Re: A simple question about Mahoma, Christ, Buda and Mathematics By that times, Theology and Mathematics were both directly related to God, moreover, ancient mathematicians and theologians were the same thing. Theology basically relates matters of divinity, the study of God and His relations to the Universe and mankind. Ancient Greeks, i.e.: Nichomacus, believed that Quantity was there long before the Creation of the World, that is, long before mankind and human feelings. In general, ancient mathematicians believed that God used Quantity as an intelligible Entity for creating the world. nor asked any single question relating those issues to Christ, Muhammad or Buddha. Ancient mathematicians were basically theologians. Domingo Gomez Morin mipagina.cantv.net/arithmetic ==== Subject: Re: A simple question about Mahoma, Christ, Buda and Mathematics <2617641.1140053213003.JavaMail.jakarta@nitrogen.mathforum.org Ancient mathematicians were basically theologians. Yes, clearly. Modern mathematicians are just as clearly gods. Who here would question the writings of Wiles on FLT, for example? Only an unbeliever would dare. Rich ==== Subject: Re: A simple question about Mahoma, Christ, Buda and Mathematics <2617641.1140053213003.JavaMail.jakarta@nitrogen.mathforum.org Ancient mathematicians were basically theologians. Yes, clearly. Modern mathematicians are just as clearly gods. Who here would question the writings of Wiles on FLT, for example? Only an unbeliever would dare. Rich ==== Subject: Re: Why do they teach Riemannian sums? It is what is known as doing mathematics as opposed to what many on this site choose to avoid out of stupidity. Many so-called proofs on this site (FCT, FLT...) have one thing in common: They establish that the poster has no concept of formal math but is quite happy to continue making a fool of itself. ==== Subject: Re: Why do they teach Riemannian sums? <27148757.1140038834939.JavaMail.jakarta@nitrogen.mathforum.org Then the problem was that you're too advanced for me :) Well, your last post was actually high quality and so you shouldn't feel bad. You pointed out that the definition you knew didn't agree with what I was saying. Focusing on ``what is the definition'' is half of the battle in studying mathematics. Half of the cranks on this newsgroup are only cranks because their definitions differ from what the rest of the world accepts. So making people specify what their words means is a good habit rather than a bad one. ==== Subject: Re: JSH: My doubts > I am an amateur who doesn't even like to call himself a mathematician, > as I usually don't consider myself to be one. > I have simple techniques which give short and powerful proofs where > people who are mathematicians ignore them or claim they are wrong. > So I now have factoring problem equations: > T = (x+y+vz)(vz-x) > where x, y and z are defined by > x^2 + xy + k_1 y^2 = k_2 z^2 > where you pick y, k_1 and k_2 and of course the equation that defines > x, y and z is either an ellipsoid or I think the term is a hyperbloid, > and finally > (2(v^2 - k_2)z + vy)^2 = ((1-4k_1)y^2+4T)v^2 + 4k_2(k_1y^2 - T). > Now I tried checking them to some extent, but all the showed me was > that as far as I knew they SHOULD work, but I am just an amateur, > dabbling with the algebra. > Putting them out there as I have and noting special things, like that > trying to generally solve them gives quintics, should reasonably spark > useful reply from the mathematical community...if it is as advertised. Why do you think this? These things might be interesting to you, an amateur who has very poor mathematical instincts and no real discernable mathematical talent, but they are not necessarily of interest to those who have seen the beauty in the field. Sorry to be harsh, but it's true. Anyone who has spent even a modest amount of time learning what has been done in the field will look at your equations, read a little of your commentary, realise you don't know what you're doing, and that your method is doomed to failure. If you want a start at understanding what's been done in the field -- and I'm sure you don't -- then you need to have some ability in number theory, probability theory, abstract algebra, and complexity theory. Once you have some understanding of the basics in these fields, then you'll know that the e that you currently call math is useless. This is why people aren't showing interest. Think about it James. If you saw the beauty in a field of mathematics, and had some true understanding of the field, would you stop to look at the scribblings of a person who clearly had no clue about the field? Maybe you would, but what if said person refused to learn anything about the field? What if the person constantly proclaimed their genius in the field, and uttered nothing but contempt toward all others in the field? Ever hear the proverb, There is no royal road to learning? You really should take this to heart. Put in the time to learn something. You'll see that there's much beauty to behold. > However, if mathematicians as a society ARE corrupt, then they can sit > without noting the equations do work, or just not notice as their > corruption has destroyed their mathematical judgement. > So we wait... > If months pass and then I simply fully implement the equations, then > there will be no debate if that implementation works, but no room to > run for mathematicians then either. > Or they won't work, and I'll simply have been wrong--an amateur who got > it wrong, yet again. > But if they do work, then all that time wasted will have been totally > the fault of a corrupt math society, when something could have been > done to protect people's investments, and nothing was because > mathematicians are a lost people. > James Harris ==== Subject: Re: JSH: My doubts <11v7m1thbr1fr80@corp.supernews.com> eh, he's probably just a victim of algebraware, which might make him a Wolframite. incidentally, you may have been quoting Hobbs, who didn't know about Bucky Fuller, then. (er, see the last URL in my sig .-) thus quoth: Ever hear the proverb, There is no royal road to learning? You really thus: Moving Dimensions Theory is *brand* new; eh?... you should read Smolin's book for a nice, low-key dyscussion of the quality of stringtheory; he's a looptheorist, though.... wow; maybe some one'd take you seriously, then. --Give Earth a Trickier Dick Cheeny -- out of office, after gigayears! http://tarpley.net/bush8.htm http://www.benfranklinbooks.com/ http://members.tripod.com/~american_almanac http://www.wlym.com/pdf/iclc/howthenation.pdf http://www.rand.org/publications/randreview/issues/rr.12.00/ http://www.rwgrayprojects.com/synergetics/plates/figs/plate01.html ==== Subject: Re: exact solution on 2d heat equation with source term yes, the period of sine function can be different for x and y. put> F = C * sin(m*x) * sin(n*y) thus, Fxx = - m^2 * C * sin(m*x)sin(m*y) Fyy = - n^2 * C * sin(m*x)sin(m*y) replace the equation, F_xx + F_yy = kF by these terms: -2 * (m^2 + n^2) * C * sin(m*x)sin(m*y) = k * C * sin(m*x)sin(m*y) k-m^2-n^2 = 0 or sin(mx)=0 or sin(ny)=0 In addition, Fxx = -m^2 * F and Fyy= -n^2 * F By general solution of Hxx + C*H =0 (C>0), H = c cos(alpha x) + d sin(alpha x) Thus, general solution of Fxx=-m^2 is F = c cos(m x) + d sin(m x) and by boundary condition F(x=0) = 0, c = 0 and F(x=1) = 0, m = s*pi (where s is an integer) Thus, F(x) = d sin(s*pi*x) Similarly, by Fyy = -n^2 F F = e cos(n x) + f sin(n x) and by boundary condition, F(y) = f sin(r*pi*y) Finally, F(x,y) = C sin(s*pi*x)*sin(r*pi*y) from G_t = kG, G = C * e^(k *t) Thus, the general solution is u=FG=C sin(s*pi*x)*sin(r*pi*y)*e^(k *t) and, the summation of general solution and particualr solution is u=C sin(s*pi*x)*sin(r*pi*y)*e^(k *t) + 1/(2*pi^2) *sin(pi*x)*sin(pi*y) Using the Initial condition, when t=0, u(x,y,0)=0. s =1 and r=1 and C = 1/(2*pi^2) thus, u = 1/(2*pi^2) * (e^(k *t) +1) * ( sin(pi*x)*sin(pi*y) It seems somewhat weird.I think I'm lost again. J.J. ==== Subject: Re: Linear Algebra and Peter Lax's book I took out the book you mentioned and it is great. It is what I was looking for. ==== Subject: How do you figure the square root of a number without a calculator? Is there any way to figure the square root of a number without a calculator? (Excluding the guess-and-check method) ==== Subject: Re: How do you figure the square root of a number without a calculator? There is an algorithm for calculating square roots that some of us learned in elementary school. It is explained in Chrystal's Textbook of Algebra. ==== Subject: Re: How do you figure the square root of a number without a calculator? Are we talking integers, rationals, forget irrationals of course (reduces to rational problem again), or complex over rationals? ==== Subject: Re: How do you figure the square root of a number without a calculator? > Is there any way to figure the square root of a number without a > calculator? (Excluding the guess-and-check method) ==== Subject: Re: Where Is My Headline? (Was Re: Continuing Failure Of Neoclassical Theory Of Value) (also presented in Chapter 15 of Mas-Colell et al). The authors note > that This model is known as the 2 X 2 production model. In > applications, factor 1 is often thought of as labor and factor 2 as > capital (pg.532, 1995 edition). This is nevertheless a two input > model, and though these labels may be convenient, the model treats > capital as a homogenous input, with a quantity that may be expressed > in physical terms (i.e. number of machines). Furthermore, production > of capital is not modeled formally, so capital is treated as an > endowment. Are the insights of such a model applicable to a world > with multiple capital goods that are produced (machines come in many > forms, often involving disparate production techniques)? The common > answer employed in these discussion is to see how the model fits the > data (i.e. use an empirical test). However, how does one construct a > meaningful measure of capital? The theory itself provides no > guidelines, as there is only type of capital good and hence no > aggregation issue. A second possibility for testing the applicability > of a one capital good model would be to work with a less restrictive > model (with multiple capital goods) and ask whether any conditions on > technology, etc. would allow one to add up the various capital goods > to form an aggregate that has the same properties as the capital good > in the two input case. My understanding of this literature is that > the necessary conditions are extremely restrictive. The stringency > of these restrictions is interpreted by some theorists to imply that > the specialized models used by applied economists have no rigorous > microfoundations, as it is not clear what restrictions on the more > general model would lead to a model with the features of the > specialized model. I should point out that I agree with the skeptical assessment about the relevance of overly aggregated models. I am partial to Franklin Fisher's quip There is no such thing as macroeconomics -- which was said with the tongue only partially in cheek (and it also applies to two factor models that pretend to say something about large economies). But then again, macroeconomics is something that became an obsession for the economic profession only with the introduction and popularization of Keynesian theories -- it really would not be such a shame if macroeconomics just went away. Maybe this will eventually mean the return to what economics looked before Keynes -- price theory on one hand and monetary economics on the other hand, with no other place for macroeconomics. (Although whether applications of game theory will become a subfield in itself is something that could be speculated upon) The online version of The Encyclopedia of Concise Economics mentions that it was Joan Robinson that was the first to introduce the definition of macroeconomics -- if they are correct, then it would be highly ironic that she was also instrumental in bringing about the downfall of macroeconomics. Alex ==== Subject: Re: Where Is My Headline? (Was Re: Continuing Failure Of Neoclassical Theory Of Value) <43ccf06e$0$25239$a8266bb1@reader.corenews.com> No. Perhaps I was unclear. > I say that Bliss and Hahn, for example, say that the Cambridge > Criticism does not affect general equilibrium theory, and it is > general equilibrium theory that, for them, is the rigorous > neoclassical theory. See the text around footnotes 5, 6, 7, and > 8 in the previously linked paper. > I am interested in ultimately attacking this position. So in other words, there is no accepted refutation of this Bliss and Hahn position, namely that the Cambridge criticism does not affect general equilibrium theory, nor do you have a refutation of it -- it is good that you agree to that. [...] > It would thus be disingenuous to > claim that those critiques somehow mean the failure of the > neoclassical theory of value, as your title suggest... > No. The mainstream response to the Cambridge capital controversy > was to drop the neoclassical theory of value. Mainstream economics > has some mathematics left, which is not the same thing. That depends on what you define to be the neoclassical theory of value -- I see no reason to accept your idiosyncratic definition of what this was supposed to be, especially since under neoclassical theory we have grouped a very wide range of research programs. I can certainly accept that at one time, hopes were high for very simple and straightforward theoretical results -- but I don't see why the complexity of the economy as revealed by general equilibrium represents a failure of neoclassical economics. Real world economics is a complex affair -- it should be no surprise that the models of economics should display such a complexity in both formulation and prediction. [...] > By explicitly making the above statement, Alex is being more honest > than my experience has been with most economists in sci.econ. > That partial equilibrium theory is just an approximation -and it > works as an approximation only under certain circumstances-is, > [...] > Alex has deleted, among other comments: > I notice Alex does not specify those circumstances. You have the habit of repeating the same point in different paragraphs -- hence I delete the repetitive paragraphs and I try to answer what I perceive to be the objection at one location; and I will continue to do this. Apparently you don't like this -- but I think that this helps clarity and I suggest that your habit is one of the reasons that you express your points so poorly. Regarding the circumstances under which PE works as an approximation of GE -- I don't think that there is a closed form expression for when this is so. I have already stated my opinion that each particular application should justify why, in the particular circumstances under study, PE --or whatever model is applied-- can be seen as a reasonable approximation of rigorous GE. Thus, even though in general no closed form expression exists for clarifying when PE is a good approximation, in particular circumstances this justification might be easier, due to the particular nature of the problem. If the particular application can provide such a justification, then it is rigorous economics -- if it can not provide it, then it is not. I will delete further references to the justification of PE applications by GE, since my answer to these is contained above. [...] > GE models formalize the dependence of prices on production functions > and subjective preferences. As such, they have much to say about > capitalist economies. > This ignores arguments about the structure of that formalism. For > example, in the previously linked paper, I say: > The critics have pointed out that the reliance on such short > period models is a major change in economic method, and > have questioned whether such models can tell us anything > about actual economies. In any approach to equilibrium in > historical time, expectations and endowments of some > capital goods would change; since expectations and endowments > are among the data of such models, an equilibrium path in these > models, as determined by the data, is of no relevance for the > actual path of actual economies. > Since I am just echoing the literature, I give references such > as the reference by Fabio Petri that Alex deleted. You can see > Petri quoting Franklin Fisher to the same purpose in the Wikipedia > entry on General Equilibrium. > When will Alex address arguments in the text supposedly under > discussion or in the broader literature? His answer probably will > be, Never. How about never? Is never a good time for you? I am not sure that I understand the objection. In GE models with sequential trade, the endowments do depend on the state or event that is realized -- and expectations also change depending on what state or event is realized. As far as I can tell, the objection is to the assumption of rational expectations -- or alternatively, to the limitation of such models to risk, without dealing with genuine uncertainty. That might be a way to go about criticizing GE but it seems like a last ditch effort -- besides the fact this does not amount to an internal criticism of GE, the rational expectation assumptions remain the best there are to deal in a systematic way with how individuals make decisions. [...] > Additionally, and contrary to what you previously claimed on this > subject, GE does make strong testable predictions about the economy -- > i.e. it is falsifiable in non-trivial ways. > Because of practical and ethical constraints, it might be extremely > difficult to test the model in its full generality. But that is simply > the cross that economists --as students of an extremely complex subject > in which it is hard to make experiments-- have to bear. > Maybe Alex should take this up with Alan Kirman (The Emperor Has > No Clothes) or perhaps, Mas-Colell. In the standard graduate text, > Mas-Colell et al. title the section on the Sonnenschein-Debreu-Mantel > results Anything Goes. In a paper referenced in the previously > linked text, Mas-Colell refers to the Capital Theory Paradoxes as > demonstrating Anything Goes. > So Alex's mere assertion about General Equilibrium's testable > predictions, as usual, is, to be nice, in tension with what the > literature argues. And Alex ignores arguments previously given on > this thread, or, rather, its predecessors (e.g., by H. Economicus). I am perfectly aware of the discussion of Sonnenschein-Debreu-Mantel results in Mas-Colell -- but unlike you, I am also aware of the fact that the interpretation of those results as implying that general equilibrium does not have testable predictions has been proved wrong. The Sonnenschein-Mantel-Debreu results _do_ _not_ imply that general equilibrium does not have empirically testable implication. This interpretation of the results has been shown to be rubbish for some time now. See for instance: Chiappori, P.-A., I. Ekeland, F. Kubler (2002) Testable implications of general equilibrium theory: a differentiable approach Working Paper No. 02-10 which is available in full here: http://www.brown.edu/Departments/Economics/Papers/2002/wp2002-10.pdf and see the references in that paper for more work on the testable implications of GE. Here are some relevant extracts from that work: Our main claim in the present paper is that this view is overly pessimistic, and that general equilibrium theory can actually generate strong testable predictions, even for large economies. The main idea is in the line of recent contributions by Brown and Matzkin (1996) and Brown and Shannon (2000), and can be summarized as follows. The Sonnenschein-Debreu-Mantel approach concentrates on the properties of excess (or market) demand as a function of prices only. However, this viewpoint is not the only possible one, and actually not the most adequate for assessing the testability of GE theory. 1 As far as testable predictions are concerned, the structure of aggregate excess demand is not the relevant issue, if only because excess demand is, in principle, not observable, except at equilibrium prices - where, by definition, it vanishes. However, prices are not the only variables that can be observed to vary. Price movements reflect fluctuations of fundamentals, and the relationship between these fundamentals and the resulting equilibrium prices is the natural object for empirical observation. One of the goals of general equilibrium theory is precisely to characterize the properties of this relationship. As it turns out, this characterization generates strong testable restrictions. [...] These results indicate that the two lines contrasted above -- the 'manifold' point of view versus the SDM excess demand approach -- generate totally different (and in a sense opposite) conclusions. How can this striking discrepancy be explained? Our interpretation emphasizes a crucial difference: in the manifold approach, individual data (initial endowments) are available, whereas only aggregate variables can be observed in the SDM setting. In other words, we understand our results as suggesting the important conclusion that ***whenever data are available at the individual level, then utility maximization generates very stringent restrictions upon observed behavior, even if the observed variables (equilibrium prices in our case) are aggregate.*** is irrelevant. Individual determinants of individual choices (such as initial endowments or individual incomes) may do just as well. So yet another of your often repeated claims, that GE has no empirical implications, bites the dust. [...] > It might be of value to do that -- use capital-theoretic paradoxes > to overthrow the GE model -- except that you've done no such thing. > I thank Alex for his agreement. Now whether, for example, Bertram > Schefold and Pierangelo Garegnani have is another matter > entirely. [I leave the above in the text only to make clear that my paragraph immediately bellow is also intended as a criticism of the attempts by Schefold and Garegnani to overthrow GE] [...] > I would think that if the structure > of the model required stability to be empirically applicable, as > some (Debreu?) think GE does, a failure to specify assumptions > that yield stability is a problem. But as we've seen above, the claims about the lack of testable prediction of GE are highly exaggerated. As such, instability is not a theoretical problem for GE, although it might be a problem for models of reality that envision a simple behavior for economic systems. Given this, the efforts of Schefold and Garegnani, to the extent that they are related to stability problems, would not form an internal critique of GE, even if they are successful in their endeavors. [...] > Somehow it must just be an oversight that Alex overlooked the following > in the post to which he is pretending to respond: > The reader might notice that Alex does not comment on what's > taught to undergraduates. I expect my reaction to Krugman's new As far as microeconomics goes, undergraduates are taught PE -- and sometimes they are taught GE in intermediate classes. I don't see what is so outrageous about this. About macroeconomics though, I tend to agree that they tend to be taught mostly nonsense. > you are just refuting a > dubious historical interpretation of what the neoclassical vision is > supposed to be. > No. Mas-Colell et al., for example, are not clear on the structure > of the theory they present. See H. Economicus' post for a > demonstration that they mistakenly call land, capital, for > example. None of this contradicts my point that you are refuting simply a dubious interpretation of what the neoclassical vision is supposed to be. GE stands unrefuted as the rigorous neoclassical economics. Alex ==== Subject: Is there an easier way to calculate the square of a number? I was recently learning how to calculate square roots and i was wondering, is there any way to calculate the square of a number without lengthy figuring? For example: in order to calculate 12^9 is there a shorter way then to do 12 x 12 x 12 x 12 x 12 x 12 x 12 x 12 x 12? ==== Subject: Re: Is there an easier way to calculate the square of a number? <24199322.1140060164076.JavaMail.jakarta@nitrogen.mathforum.org>, > I was recently learning how to calculate square roots and i was wondering, is > there any way to calculate the square of a number without lengthy figuring? > For example: in order to calculate 12^9 is there a shorter way then to do 12 > x 12 x 12 x 12 x 12 x 12 x 12 x 12 x 12? That's a square? Yes, there is a quicker way: 12 x 12 = 144 (= 12^2) 144 x 144 = 20736 (= 12^4) 20736 x 20736 = 429981696 (= 12^8) 429981696 x 12 = 5159780352 (= 12^9) To compute 12^n, break n down into binary, and use the repeated squaring trick above. In the usual multiplication algorithm, this requires (if I've not miscounted) 56 one-digit by one-digit multiplies. The straightforward way of computing 12*12*... seems to take 88. Plus a whole bunch of additions, which of course are where most of the work is. --Ron Bruck ==== Subject: Re: Is there an easier way to calculate the square of a number? days. My association with the Department is that of an alumnus. >I was recently learning how to calculate square roots and i was wondering, is there any way to calculate the square of a number without lengthy figuring? For example: in order to calculate 12^9 is there a shorter way then to do 12 x 12 x 12 x 12 x 12 x 12 x 12 x 12 x 12? For one thing, you can square. (12)^9 = (((12)^2)^2)^2 * 12. This requires only four multiplications. -- ' ==== Subject: Re: Is there an easier way to calculate the square of a number? > I was recently learning how to calculate square roots and i was > wondering, is there any way to calculate the square of a number without > lengthy figuring? For example: in order to calculate 12^9 is there a > shorter way then to do 12 x 12 x 12 x 12 x 12 x 12 x 12 x 12 x 12? Use binomial theorem 12^9 = (10 + 2)^9 = 10^9 + 9*10^8 * 2 + 9*8/2 * 10^7 * 2^2 + ... or 12^9 = 3^9 * 2^9 * 2^9 2^9 = (1 + 1)^9 = 9 + 9*8/2 + ... 3^9 = (2 + 1)^9 = 2^9 + 9*2^8 + ... or 12^9 = (6 + 6)^9 = 6^9 (9 + 9*8/2 + ...) 6^9 = (3 + 3)^9 = 3^9 (9 + 9*8/2 + ...) 3^9 = (2 + 1)^9 = 2^9 + 9*2^8 + ... ==== Subject: Re: Is there an easier way to calculate the square of a number? >I was recently learning how to calculate square roots and i was wondering, >is there any way to calculate the square of a number without lengthy >figuring? For example: in order to calculate 12^9 is there a shorter way >then to do 12 x 12 x 12 x 12 x 12 x 12 x 12 x 12 x 12? ==== Subject: Re: Ramsey's Theorem/Graph Theory Question > Show that every graph with n vertices contains either a clique or an > independent set with (1/2)log[base 2] n vertices. >> I'm not clear on the terminology being used. Independent set -- >> meaning a set of edges with no cycles? If so, the claim fails for a >> discrete graph (a graph with no edges) of 3 or more vertices. And does >> clique mean a set of three pairwise adjacent vertices? That was >> called a 3-clique back in the disco era, when I last looked at this >> stuff :) > If you're not going to help, don't post. > An independent set is a set of vertices, where no two are adjacent. > A clique is a set of vertices, where every pair is adjacent. > Now onto the proof. You should probably try induction, where the > induction step consists of taking a graph and partitioning it into two > (nearly) equal sizes. > --- Christopher Heckman log[base 2] n? Then show that it works for base case of n = 1, etc... Nate -- The life of a repoman is always intense! ==== Subject: Re: linear constant coefficient nonhomogenous difference equation > Hi. > I wonder if there exists an algorithm or an exact formula > to solve the following equation for all kinds of f(n) > and constant coefficients > y(n+m)+a_1 y(n+m-1)+...+a_m y(n)=f(n) > if you have any information, or even you know that such thing doesn't exists please tell me. Here is how you can do it. For illustrative purposes, I'll deal in detail with the case of m=1, so the recursion is ay(n) + y(n+1) = f(n). Now let's take the moment generation function (mgf) of both sides. The mgf of the sequence {f(n)} is F(z) =def= sum{z^n f(n): n=0 .. infinity}, while Y(z) =def= sum{ z^n y(n): n=0..infinity} is the mgf of {y(n)}. Now we have F(z) = a Y(z) + sum{z^n y(n+1):n=0..infinity} = a Y(z) + (1/z) sum{ z^n y(n): n=1..infinity} = aY(z) + (1/z) [ Y(z) - y(0)]. Therefore, Y(z) = [F(z) + (1/z) y(0)]/[a + (1/z)] =[y(0) + zF(z)]/[1 + az] = y(0)/(1+az) + [z/(1+az)]F(z). Now 1/(1+az) = sum{(-a)^n z^n: n=0..infinity} and g(z) =def= z/(1+az) = sum{(-a)^(n-1) z^n : n=1..infinity}. This is the mgf of the sequence {g(n)}, where g(0) = 0 and g(n) = (-a)^(n-1) for n >=1. The product g(z)F(z) is the mgf of the convolution of the sequences {g(n)} and {f(n)}, so the coefficient of z^n in this product is sum{ g(k) f(n-k): k=0..n}. Altogether, the coefficient of z^n in Y(z) is y(n) = y(0)(-a)^n + sum{ (-a)^(k-1) f(n-k): k=1..n} (since g(0) = 0). (You should check my algebra, just to be sure.) Now, for a more general case such as f(n) = a1y(n) + a2y(n+1) + y(n+2) you would get Y(z) = {z^2 F(z) + y(0) + z[y(0) + y(1)]}/{1+a2 z + a1 z^2}. By factoring the denominator, you can express the answer in terms of convolutions of {f(n)} and geometric series in the roots of the denominator. For even higher values of m the same thing can be done, but it gets increasingly messy. Good luck R.G. Vickson Adjunct Professor, University of Waterloo ==== Subject: Re: The good Christian should beware of mathematicians <8fadndv_E8pm2XDeRVn-iA@hamilton.edu> I'd like to add that Augustine wasn't completely adverse to using Greek background. (E.g. when he was young, he rejected Christianity because of the popular belief of a corporeal God, and when he could reconcile the Bible with Plato, he came back.) That's good reason to think that he was rejecting a particular class of Greek thinkers, especially given his emphasis on the use of prophecy. ==== Subject: Re: More or Less Than the Sum of Its Parts <43d64dce.45572551@netnews.att.net> <43d6a794.55945980@netnews.att.net> <43d7d9d0.70677259@netnews.att.net> <43daa171.7085064@netnews.att.net> <20060130164619.558$Zb@newsreader.com> <43df91b5.61423506@netnews.att.net> <20060206160212.036$Ao@newsreader.com> <43e7f9d7.54656180@netnews.att.net> <20060206225139.156$9K@newsreader.com> <43e8ee01.60915852@netnews.att.net> <20060207143858.314$7h@newsreader.com> But you need each other to argue with, to structure time. ==== Subject: Re: More or Less Than the Sum of Its Parts <43d64dce.45572551@netnews.att.net> <43d6a794.55945980@netnews.att.net> <43d7d9d0.70677259@netnews.att.net> <43daa171.7085064@netnews.att.net> <20060130164619.558$Zb@newsreader.com> <43df91b5.61423506@netnews.att.net> <20060206160212.036$Ao@newsreader.com> <43e7f9d7.54656180@netnews.att.net> <20060206225139.156$9K@newsreader.com> <43e8ee01.60915852@netnews.att.net> <20060207143858.314$7h@newsreader.com But you need each other to argue with, to structure time. Like a clock ? ==== Subject: Quadratic residues Given a positive integer n, must there be a prime p such that n is a quadratic residue modulo p? Infinitely many such primes? The first picture at http://mathworld.wolfram.com/QuadraticResidue.html suggests that the answer to each question is yes. It looks like every row is going to have infinitely many dots. [n is a quadratic residue modulo p means there exists an integer x with 0 < x < p and x^2 = n (mod p).] ==== Subject: Re: proving Kepler Packing and 4 Color Mapping using the same technique of Smallest Member Nevada is surrounded by ... Arkansas? thu qouth: > Example: Nevada is surrounded by Ca,Or,Id,Ut,Ar and is not 4 mutual thus: eh, he's probably just a victim of algebraware, which might make him a Wolframite.... incidentally, you may have been quoting Hobbs, his no royal road to geometry, who didn't know about Bucky Fuller, then. (er, see the last URL in my sig .-) thus: Moving Dimensions Theory is *brand* new; eh?... you should read Smolin's book for a nice, low-key dyscussion of the quality of stringtheory; he's a looptheorist, though.... wow; maybe some one'd take you seriously, then. --Give Earth a Trickier Dick Cheeny -- out of office, after gigayears! http://tarpley.net/bush8.htm http://www.benfranklinbooks.com/ http://members.tripod.com/~american_almanac http://www.wlym.com/pdf/iclc/howthenation.pdf http://www.rand.org/publications/randreview/issues/rr.12.00/ http://www.rwgrayprojects.com/synergetics/plates/figs/plate01.html ==== Subject: Re: JSH: My patience has ended well, your point is taken, since JSH hasn't followed-up even once. or, perhaps, he was canoodling with his groupies for V-day. thus: eh, he's probably just a victim of algebraware, which might make him a Wolframite.... incidentally, you may have been quoting Hobbs, his no royal road to geometry, who didn't know about Bucky Fuller, then. (er, see the last URL in my sig .-) thus: Moving Dimensions Theory is *brand* new; eh?... you should read Smolin's book for a nice, low-key dyscussion of the quality of stringtheory; he's a looptheorist, though.... wow; maybe some one'd take you seriously, then. --Give Earth a Trickier Dick Cheeny -- out of office, after gigayears! http://tarpley.net/bush8.htm http://www.benfranklinbooks.com/ http://members.tripod.com/~american_almanac http://www.wlym.com/pdf/iclc/howthenation.pdf http://www.rand.org/publications/randreview/issues/rr.12.00/ http://www.rwgrayprojects.com/synergetics/plates/figs/plate01.html ==== Subject: Re: Cantorian pseudomathematics <271d5$43f19e0c$82a1e228$5729@news1.tudelft.nl> <878xsdatgb.fsf@phiwumbda.org> <25ba$43f31182$82a1e228$20285@news2.tudelft.nl> statement without observable implications been later found to be of any > practical value, What about abstract algebra? Finite fields of order 2^n for example? Are they with or without observable implications? Were they with or without observable implications in Hardy's time? They have certainly later been found to have (immense) practical value, contrary to what Hardy thought - how would the observable implications be assessed? In one interpretation your claim is supported, because it can be reduced to: Never in the history of m&s has a statement not later being found to be of any practical value, later been found to be of any practical value. > ... other than as a weapon of use in psychological warfare. Well, if you weren't so keen on writing lunatic nonsense like this, it might help people to consider what ideas you might have. Brian Chandler http://imaginatorium.org ==== Subject: Re: Cantorian pseudomathematics > OK. I will use a language that even the most stubborn > mathematician can understand. > > Consider the following function: > > sqrt(t^2 + 9) - 3 > g(x) = ----------------- for x <> 0 > t^2 > = 1 for x = 0 > > Definition: a function f is _sensible_ at a iff > > lim f(x) = f(a) > x->a > > Definition: a function f is _nonsense_ at a iff > it is not sensible at a. > > Prove that the above function g is nonsense at 0 . > BTW, mathematicians already have a concept of what you said above, it's > called continuous, > That's not exactly what Han saying. Like many constructivists, he's > arguing that it doesn't make sense to say that a function has a value > at a point of discontinuity. I thought constructionists required all functions to be contiuous. ==== Subject: Re: Cantorian pseudomathematics <87psm0g362.fsf@phiwumbda.org> <271d5$43f19e0c$82a1e228$5729@news1.tudelft.nl> <878xsdatgb.fsf@phiwumbda.org> <25ba$43f31182$82a1e228$20285@news2.tudelft.nl> <45hpgqF6pg29U2@individual.net The Cantorians are conducting a campaign of psychological warfare > against humanity. > That is nonsense. They are just playing with ideas. Not true. When they call anyone who questions their ideas a crackpot, they are doing something more than just playing. ==== Subject: Re: Cantorian pseudomathematics > The Cantorians are conducting a campaign of psychological warfare > against humanity. > That is nonsense. They are just playing with ideas. > Not true. When they call anyone who questions their ideas a crackpot, > they are doing something more than just playing. Then so are those who call their ideas crackpot, which is what most of them are objecting to. Cantorians by and large do not wish to prevent others from thinking their own non-Cantorian thoughts, but violently resent being told that they should be penalized for or prohibited from thinking Cantorian thoughts. Such though police activity by those vociferous anti-Cantorians is Fascist. ==== Subject: Re: Cantorian pseudomathematics <271d5$43f19e0c$82a1e228$5729@news1.tudelft.nl> <878xsdatgb.fsf@phiwumbda.org> <25ba$43f31182$82a1e228$20285@news2.tudelft.nl> <45hpgqF6pg29U2@individual.net> I think that's ridiculous. Let's talk about truth in ZF. The only true thing about each thing in ZF is that it's a set. So, then you try and make a set of all the sets and it's not a set, but that predicate for any argument evalutes to true as there is quantification over the universe, a universal quantifier. Then it's not true that everything in ZF is a set, but that means the axioms are wrong, false. That each and every thing is a set, and that everything is a set, are in a way mutually implicative statements. Either means the other. So, you are perhaps barking up the wrong tree as you mark like bad, but it is rather inescapable that there is a universe, and there's not one in ZF, so, ZF is not any kind of adequate foundation for any and all mathematical objects. Where there is a universe, a universal set, it's its own powerset. Is that a paradox? No, the answer to whether that is a paradox is no. It exists in naive set theory, and a variety of antifoundational set theories, which in terms of discussion of the melange of all possible mathematical objects, within the universe or domain of discourse, is perhaps counternomenclaturally a BETTER foundation, more fundamentally sound and able to make sense of these things that _must_ be, than the current state of axiomatized foundations. Again, solid foundations are emergent from the null axiom theory. There is no universe in ZF. There's no universe in ZF with classes or any other extension of the group game. Because of the axiomatized regularity of sets, even where empty set membership means regularity in a wide variety of the axioms in use that nobody ever actually really uses throughout the extension of logic, only in reference, because of the axiomatized regularity of sets ZF and simliar regular theories are inconsistent. ZF is inconsistent. It's definitely incomplete, that there are true statements in ZF that are not provable in ZF is widely agreed, and that is as well a consequence of the axiomatization, and it's inconsistent, because a predicate can never always evaluate to true, not even true. Please stop being so ridiculous. The universe is infinite. Infinite sets are equivalent. Ross ==== Subject: Re: Cantorian pseudomathematics <271d5$43f19e0c$82a1e228$5729@news1.tudelft.nl> <878xsdatgb.fsf@phiwumbda.org> <25ba$43f31182$82a1e228$20285@news2.tudelft.nl> practical (i.e. real world) value, it must have observable > implications. > But, as the history of science has often shown, what is not observable > today may be observable tomorrow, Any reasonable notion of observable will imply that the observable mathematical universe is countable. Cantor's argument implies that there exists something beyond that, which is not observable. ==== Subject: Re: Cantorian pseudomathematics > If mathematics is to have any > practical (i.e. real world) value, it must have observable > implications. > But, as the history of science has often shown, what is not observable > today may be observable tomorrow, > Any reasonable notion of observable will imply that the observable > mathematical universe is countable. Not so! It only limits us to countably many observations of what may well be an uncountable universe. Cantor's argument implies that > there exists something beyond that, which is not observable. Cantor does not argue anything about the nature of the physical (observable?) universe, but only about the universe of thought. Or doews David insist that we not be allowed to think about anything that we cannot 'observe'? ==== Subject: Re: Cantorian pseudomathematics <87psm0g362.fsf@phiwumbda.org> <271d5$43f19e0c$82a1e228$5729@news1.tudelft.nl> arguing that it doesn't make sense to say that a function has a value > at a point of discontinuity. > So there's something nonsensible about the function > f(x) = {0, x < 0 > {1, x >= 0 That's what Brouwer and many other constructivists would say. ==== Subject: Re: Observability of dimension is restricted. Paradox is a naturally occuring phenomena. Trying to measure something that is infinitely long is like trying to measure something with a one ended ruler. It just dont work. I dont think that's a Zeno paradox. ==== Subject: Re: Observability of dimension is restricted. Paradox is a naturally occuring phenomena. %IW48mQf3K=Ci&gZ7]]aazx@]Y-nq!r5{yH/#,?@lDdUDvOfByB2hVW0.@OM%{l/{cT'{w > Trying to measure something that is infinitely long is like trying to > measure something with a one ended ruler. It just dont work. That's why it's important to always use the correct tool for the job. To measure something which is infinitely long, you need an infinite ruler. -- Curt Welch http://CurtWelch.Com/ curt@kcwc.com http://NewsReader.Com/ ==== Subject: Category theory: natural transformation from functor application Let C1, C2, D and E be (small) categories. I write [D,E] for the category of functors from D to E. Let F1 and F2 be functors from C1 to [D,E], and G1 and G2 be functors from C2 to D. Let s:F1->F2 and t:G1->G2 be natural transformations. One can define functors H1 and H2 from the product category C1*C2 to E by H1 = apply_D,E o (F1*G1) H2 = apply_D,E o (F2*G2) Using s and t, is there a natural way to define a natural transformation u:H1->H2? ==== Subject: Re: Factoraization of n! as power of primes days. My association with the Department is that of an alumnus. >Hi everyone, >I wanna ask if anyone know where i can get a formula to the >factoraization of n! as a product of powers of primes i.e. n!=p1^e1 * >p2^e2 *.... where e1,e2,e3 are nonnegative integers and the formula >will provide them. There is no simple factorization. >can anyone tell me in which book i can get a formula like this ??? Your dreams? >another request can anyone tell me about that prize given to the >mathematician who's below 40 years old ? Lots of prizes are given to mathematicians; some of them are given to mathematicians who are under 40. Very few prizes have official rules specifying age; in fact, no prize that I know of. UNOFFICIALLY, the Fields Medal is given only to mathematicians who are still under forty, and I am quite sure this is the one you mean, though. -- ' ==== Subject: Re: New Conception of Time: Moving Dimensions Theory ==== Subject: does midnight belong to (open) or [closed] interval,(day endg] or [day beginning?)HELP.. max's curious questions greetings. (0,24) includes neither endpoint. [0,24] both (0,24] closed on right [0,24) closed on left. (a,b) parentheses [c,d] brackets. this is thoroughly?? covered in university mathematics analysis limits?? which day? does midnight belong to.. the day ending or the day starting? neither or both. in 99.99 % of cases it is ambiguous please yourself or undefined. THErefore very bad communication. thx gar.. cc.. ATTN simon Beattie WCC communication. taps turned off??? and WAS WGTN ASTRON SOC bcc; christmas eve service at midnight ? end of day. hhhhhhhhhh eeeeeeeee lllllllll ppppppppp. [(HELP.)] don.mcdonald. thank you. 2 entries found for open interval. open interval n. A set of numbers consisting of all the numbers between a pair of given numbers but not including the endpoints. [Download Now or Buy the Book] Source: The American Heritage¬ Dictionary of the English Language, Fourth Edition Copyright © 2000 by Houghton Mifflin Company. Published by Houghton Mifflin Company. All rights reserved. open interval n : an interval that does not include its endpoints [syn: unbounded interval] [ant: closed interval] ==== Subject: Re: does midnight belong to (open) or [closed] interval,(day endg] or [day beginning?)HELP.. >I believe that in most western cultures, Midnight belongs same day as >the noon that preceeded it. In North America, and AFAIK all of Europe, midnight is the beginning of a new day. There is no time 24:00. It is either 00:00, or 12:00 am, and a new date. ______________________ Chris L Peterson Cloudbait Observatory http://www.cloudbait.com ==== Subject: Inexact differentials Does anybody have experience with inexact differentials. They are being used in thermodynamics. I like to use them in pottery. I would appreciate an ==== Subject: 2nd IMT-GT Conference on Mathematics, Statistics and Their Applications INTRODUCTION. Progress in the mathematical sciences within the IMT-GT region will benefit greatly from regular meetings and interaction amongst mathematicians and scientists in the region.The first IMT-GT Conference on Mathematics, Statistics and Their Applications was held in June 2005 in Lake Toba, Sumatera, Indonesia. The conference was organized by Universitas Sumatera Utara. The second IMT-GT Conference will be organized by Universiti Sains Malaysia in collaboration with The Malaysian Mathematical Sciences Society and will be held at The Gurney Hotel in Penang from June 13 - 15, 2006. The scientific program will include plenary addresses, regular presentations and workshops. OBJECTIVE. To provide a forum for researchers, educators, and students to exchange ideas, to communicate and discuss research findings and new advances in mathematics. To explore possible avenues to foster academic and student exchange, as well as scientific activities within the region. CONFERENCE TOPICS. Mathematics, Applications of Mathematics, Statistics, Operations Research, Mathematics Education and Computer Science. PLENARY SPEAKERS AND WORKSOPS. The plenary speakers and the titles of their talks are as follows: * Dr. Andonowati (Indonesia): Extreme Waves * Dr. Gan Eng Hui (Malaysia): Quality Control in Manufacturing Industries * Prof. Dr. E. Van Groesen (Netherlands): Mathematical Modelling of Tsunamis * Prof. Dr. Wolfgang H.8ardle (Germany): A Topic in Statistics * Dato' Prof. Dr. Kamel Ariffin Mohd. Atan (Malaysia): On the Explicit Estimation of Exponential Sums * Prof. Dr. I-Ming Tang (Thailand): Mathematica Modelling of Transmission of Avian Influenza Virus * Prof. Dr. Vu Kim Tuan (USA): Irregular Sampling of Bandlimited & Positive Frequency Signals There will also be workshops on Bioinformatics, Graphing Calculators, Quality Control and Generation of Tsunamis. CALL FOR PAPERS. Papers to be considered for presentation and for publication in the conference proceedings must be submitted before February 15, 2006. The proceedings will be typeset using LaTeX. To expedite publication of the proceedings, participants who are interested to present papers are encouraged to prepare their manuscripts using LaTeX (amsproc style). Papers prepared in MS Word are also acceptable, and must be set on A4 paper size with an 11-point body text. Please note that there is an 8-page limit on the length of the paper. Authors can download the guidelines and templates in the preparation of their manuscripts. A soft copy of the paper template is also available from the Conference Secretariat. Participants who are only interested to present talks are requested to submit their abstracts as early as possible before March 30, 2006. A booklet of abstracts will be made available during the conference. Abstracts of talks and full papers may be submitted via regular mail to the Conference Secretariat or via email to vravi@cs.usm.my. REGISTRATION FEE. The fee is RM400 or USD110 and must be paid before 15 April 2006. For students, the fee is RM320 or USD85 (an acknowledgement from the Head of Department that the participant is a student is required). The fee is inclusive of relevant conference materials, conference proceedings, daily lunch and tea breaks. ACCOMMODATION. The Gurney Hotel offers a special conference rate of RM150 nett for a single room and RM160 nett for a twin sharing room (inclusive of buffet breakfast). Please contact the hotel for reservations at gurneyhotel@myjaring.net (Tel: + 604 - 370 7000, Fax: + 604-370 5000). VISA. Please visit the Immigration Department of Malaysia web site for Visa regulations and forms. CONFERENCE SECRETARIAT IMT-GT RCMSA06 School of Mathematical Sciences Universiti Sains Malaysia 11800 USM Penang Malaysia E-mail: imtgt2006@usm.my, imtgt2006@yahoo.com, vravi@cs.usm.my Telephone: +60 4 653 3284/3966 Fax: +60 4 6570910 ==== Subject: Ann: ESBUnitConv v5.1 - our Free Unit Conversion Utility for Windows ESB Consultancy is pleased to announce that v5.1 of ESBUnitConv, our Free Unit Conversion Utility for Windows, has now been released. With this release we have added more customisations, including being able to toggle Prefix Functions and Precedence in the Popup Calculator. We also now have 464 conversions in 20 different measurement categories. http://www.esbcalc.com/esbunitconv.htm Glenn Crouch mailto:glenn@esbconsult.com ESB Consultancy http://www.esbconsult.com Home of ESBPCS & ESB Calculators Kalgoorlie-Boulder, Western Australia ==== Subject: Ann: ESBCalc 5.1 - our Free Calculator for Windows ESB Consultancy is pleased to announce the release of ESBCalc v5.1 - our Free Calculator for Windows. This release includes the option to turn off precedence if you would prefer 5 + 2 * 3 = 21 rather than 11. Also includes other minor improvements :) http://www.esbcalc.com/esbcalc.htm Glenn Crouch mailto:glenn@esbconsult.com ESB Consultancy http://www.esbconsult.com Home of ESBPCS & ESB Calculators Kalgoorlie-Boulder, Western Australia ==== Subject: Ann: ESBPDF Analysis v2.3 released - Probability Distribution Software for Windows The following is an Australian product :) ESB Consultancy is pleased to announce the release of ESBPDF Analysis v2.3 - our Probability Distribution Software for Windows. ESBPDF Analysis aims at making Probability Distributions easy - http://www.esbpdf.com This is a free upgrade for Registered Customers :) This latest release adds more customisation such as Trimming Trailing Zeroes and the option of Prefix Functions in the popup calculators. ESBPDF Analysis provides an easy-to-use tool for using Discrete and Continuous Probability Distributions in a single application. Most Tables and supplied functions (such as in MS Excel) give P (X < A) and using algebra other results can be found whereas ESBPDF Analysis is Probability Analysis Software that handles all the combinations for you. Features include Binomial, Poisson, Hypergeometric, Normal, Exponential, Student t, Chi Squared, F, Beta and Lognormal Distributions; Inverses of Normal, Student t, Chi Squared, F, Beta and Lognormal Distributions; Lists of Binomial Coefficients, Factorials and Permutations; Calculations of Gamma and Beta Functions; Printing of Standard Normal Tables and Critical t Values; Fully Customisable; Inbuilt Scientific Calculator; Integrated Help System which includes a Tutorial. Ideal for the Maths/Stats Student who wishes to understand Probability Distributions better, as well as the Maths/Stats Buff who wants a well designed calculating tool. Also of use for people involved in Financial and Economical modelling. Glenn Crouch mailto:glenn@esbconsult.com ESB Consultancy http://www.esbconsult.com Home of ESBPCS & ESB Calculators Kalgoorlie-Boulder, Western Australia === ==== Subject: What is the solution? Hi all you gurus, I have at little equation that I can't solve: (1+x)sqrt(1-x*x)=1 -1 < x < 1 The should be two real solutions. One is easy seen as x=0 - but what is the other? And how is it solved? Lasse ==== Subject: Re: What is the solution? > Hi all you gurus, > I have at little equation that I can't solve: > (1+x)sqrt(1-x*x)=1 -1 < x < 1 > The should be two real solutions. One is easy seen as x=0 - but what is the > other? And how is it solved? Square both sides: (1 + x)^2*(1 - x^2) = 1 Expand: 1 + 2x - 2x^3 - x^4 = 1 2x - 2x^3 - x^4 = 0 2 - 2x^2 - x^3 = 0 This cubic can be solved for its real root. ==== Subject: Function needed I have a few situations where a function starts out approximately linear - say y = 5x + 9, but y never goes past some arbitrary maximum, say 93. The behaviour when x gets larger should be sensible and continuous, say y = 83 and 92 when x = 15 and 20, in the example. I have tried to adapt some functions I know of that show asymptotic behaviour, but have not been successful. I wonder if anyone could point me in the right direction.