mm-4699 === Subject: dual construction to Kan extension? posting-account=_U59GwkAAABTw5lCldoNszkYbdR3_fpo Gecko/20080512 Camino/1.6.1 (like Firefox/2.0.0.14),gzip(gfe),gzip(gfe) Let F:A --> B be a functor and let D a category. Then there are induced functors: F^*: func(B,D) -- > func(A,D) F_* : func(D,A) --> func(D,B) Is there some nice relation between the adjoint functors of F^* and those of F_*? (assume that all adjoints exist as required) dan === Subject: Re: A beginner's question on permutation groups posting-account=mV9EXQoAAACmCMM9qg0N4eJlXyr2Z93U CLR 1.1.4322),gzip(gfe),gzip(gfe) Let H be a set containing n distinct elements. Let G be the set of all possible permutations of the n elements. æThe > elements of G form a group whose group operation is the composition of > permutation. What about the underlying set H? æDo we have a general name for it, or > role in the theory? H is called a G-set because G operates on it (here by definition). > Of course any n-element set H' can be made a G-set (i.e. an operation > of > G on H' can be defined) by fixing a bijection f:H->H', so there is > nothing too special about H after all even though it is used in the > very definition of G. In other words, the group G of permutaions of H > is isomorphic to the group G' of permutationsof H'. If you want to get confused, let H'=H and let f be an element of G. > Then of course G' = G, but which isomorphism G->G is defined by f? I dunno... an automorphism? Nothing at all? As you say, there is nothing very special about H. In fact (warming to the logical trivia) I'm not even sure we want a particular role for a particular set H... maybe the group operation is on the class of all ordered sets of n elements (maybe ordered or unique elements is either redundant or unnecessary given the nature of sets?). In that case... Hmm... doesn't seem quite right, either, because if we want a group element which maps (1,2,3,4,5) to (2,1,3,4,5), and also, one which maps (1,2,3,4,5) to 1,2,3,5,4), we don't necessarily want to say that these are the same group element, because, after all, we could map the labels (1,2,3,4,5) to (4,5,3,1,2) without really changing anything. Even if in some sense these group elements are equivalent, since beyond the distinctness of labels the original ordinering doesn't really play a role, in a particular accounting we want to keep them separate. You're right... I'm confused. How about we consider the n-permutation group, which doesn't give a figo about the labels of the underlying set -- except some imagined ordered labeling to keep the distinct operations distinct -- and then something like the n-permutation group over H, which requires us to label the elements of H: 1,...,n, and thereafter label the n! permutations in some fixed order... which still gives us only one object, gosh darn it... but secondarily labels the elements of H by Mary,Tom,Jake..., or confusingly, even 1,2,3... -- now in a second slot, so that for a fixed H (taken as identical with our second fixeded group of n distinct labels), we have n! distinct n- permutations over H, depending on the relation of our forced ordering lables 1,2,3,... to the set's original labels, Mary, Tom, etc. Here's my resolution: In some sense, there is only one n-permutation group. In another, there are an indefinite number of them, because there are an indefinite number of labels possible for the underlying set H. For each of these arbitrary distinct choices of labelings of H, there are further n! variations for some standard ordering of the group operations. Or, if we dispense with a standard ordering then each distinct set of labels of H gives just one distinct version of the group (Tom,Mary --> Mary,Tom is a single element regardless of whether Tom is originally in slot 1 and Mary in slot 2, if we don't place labels on the slots). To answer your last question, taking a bijection of H into H, that either changes nothing in the order-free listing of group operations, or else takes us into another of the n! variations for that set of labels H, assuming the bijection leaves our imposed ordering intact (Tom and Mary switch numbered seats before we start permutating), or else the result in undefined. Or maybe, even numbering the seats, we merely start at another point in the group's graph, possibly relabeling that vertex as 1, and possibly not caring. And I just nodded my head at the permutation group! before! === Subject: Re: A beginner's question on permutation groups Let H be a set containing n distinct elements. Let G be the set of all possible permutations of the n elements. The > elements of G form a group whose group operation is the composition of > permutation. What about the underlying set H? Do we have a general name for it, or > role in the theory? H is called a G-set because G operates on it (here by definition). > Of course any n-element set H' can be made a G-set (i.e. an operation > of > G on H' can be defined) by fixing a bijection f:H->H', so there is > nothing too special about H after all even though it is used in the > very definition of G. In other words, the group G of permutaions of H > is isomorphic to the group G' of permutationsof H'. If you want to get confused, let H'=H and let f be an element of G. > Then of course G' = G, but which isomorphism G->G is defined by f? I dunno... an automorphism? Nothing at all? As you say, there is nothing very special about H. In fact (warming to the logical trivia) I'm not even sure we want a particular role for a particular set H... maybe the group operation is on the class of all ordered sets of n elements (maybe ordered or unique elements is either redundant or unnecessary given the nature of sets?). In that case... Hmm... doesn't seem quite right, either, because if we want a group element which maps (1,2,3,4,5) to (2,1,3,4,5), and also, one which maps (1,2,3,4,5) to 1,2,3,5,4), we don't necessarily want to say that these are the same group element, because, after all, we could map the labels (1,2,3,4,5) to (4,5,3,1,2) without really changing anything. Even if in some sense these group elements are equivalent, since beyond the distinctness of labels the original ordinering doesn't really play a role, in a particular accounting we want to keep them separate. You're right... I'm confused. How about we consider the n-permutation group, which doesn't give a figo about the labels of the underlying set -- except some imagined ordered labeling to keep the distinct operations distinct -- and then something like the n-permutation group over H, which requires us to label the elements of H: 1,...,n, and thereafter label the n! permutations in some fixed order... which _still_ gives us only one object, gosh darn it... but secondarily labels the elements of H by Mary,Tom,Jake..., or confusingly, even 1,2,3... -- now in a second slot, so that for a fixed H (taken as identical with our second fixeded group of n distinct labels), we have n! distinct n- permutations over H, depending on the relation of our forced ordering lables 1,2,3,... to the set's original labels, Mary, Tom, etc. Here's my resolution: In some sense, there is only one n-permutation group. In another, there are an indefinite number of them, because there are an indefinite number of labels possible for the underlying set H. ********************** In that sense, we have an infinite number of groups C2, because we can lable the elements as 0, 1 or red, blue, or plus,minus or any of another infinite sets of lables. There is, however, only one group C2. For each of these arbitrary distinct choices of labelings of H, there are further n! variations for some standard ordering of the group operations. Or, if we dispense with a standard ordering then each distinct set of labels of H gives just one distinct version of the group (Tom,Mary --> Mary,Tom is a single element regardless of whether Tom is originally in slot 1 and Mary in slot 2, if we don't place labels on the slots). To answer your last question, taking a bijection of H into H, that either changes nothing in the order-free listing of group operations, or else takes us into another of the n! variations for that set of labels H, assuming the bijection leaves our imposed ordering intact (Tom and Mary switch numbered seats before we start permutating), or else the result in undefined. Or maybe, even numbering the seats, we merely start at another point in the group's graph, possibly relabeling that vertex as 1, and possibly not caring. And I just nodded my head at the permutation group! before! ****************** You are making this far more complicated than it needs be. There is only one permutation group Sn; changing the names of the elements without changing the group structure does not create another group; they are all isomorphic. === Subject: What's the good word? posting-account=mV9EXQoAAACmCMM9qg0N4eJlXyr2Z93U CLR 1.1.4322),gzip(gfe),gzip(gfe) If I have eight objects in slots labeled 1,2,...,8, then I say the objects are ordered. If I have the same eight objects inserted in slots labeled Tom, Fred,....,Haggis,Horton, then I've imposed some order on them -- but I have not implied any ordering relation as inherent in the integers. What, if anything, is the shorthand for this idea? Bijectively mapped to a second set of equal cardinality doesn't seem catchy. === Subject: Re: What's the good word? posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > If I have eight objects in slots labeled 1,2,...,8, then I say the > objects are ordered. If I have the same eight objects inserted in slots labeled Tom, > Fred,....,Haggis,Horton, then I've imposed some order on them -- but > I have not implied any ordering relation as inherent in the > integers. æWhat, if anything, is the shorthand for this idea? What about (3 objects for simplicity): P = { 2, { 1, { 0 } } } N = { Tom, Jack, Fred } Bijectively mapped to a second set of equal cardinality doesn't seem > catchy. I wouldn't know... It seems to me the numbers above are still just names. Anyway, P shows a structure that N lacks. BTW, in relational calculus we have relations and eventually their indexes... -LV === Subject: Re: What's the good word? posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) If I have eight objects in slots labeled 1,2,...,8, then I say the > objects are ordered. If I have the same eight objects inserted in slots labeled Tom, > Fred,....,Haggis,Horton, then I've imposed some order on them -- but > I have not implied any ordering relation as inherent in the > integers. æWhat, if anything, is the shorthand for this idea? What about (3 objects for simplicity): P = { 2, { 1, { 0 } } } N = { Tom, Jack, Fred } How about an indexing: N = { Tom, Jack, Fred} P = { , , } = = { Gamma, { Beta, { Alpha } } } -LV Bijectively mapped to a second set of equal cardinality doesn't seem > catchy. I wouldn't know... It seems to me the numbers above are still just > names. Anyway, P shows a structure that N lacks. BTW, in relational calculus we have relations and eventually their > indexes... -LV === Subject: Re: What's the good word? posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > If I have eight objects in slots labeled 1,2,...,8, then I say the > objects are ordered. If I have the same eight objects inserted in slots labeled Tom, > Fred,....,Haggis,Horton, then I've imposed some order on them -- but > I have not implied any ordering relation as inherent in the > integers. æWhat, if anything, is the shorthand for this idea? What about (3 objects for simplicity): P = { 2, { 1, { 0 } } } N = { Tom, Jack, Fred } > How about an indexing: N = { Tom, Jack, Fred} P = { , , } = > æ = { Gamma, { Beta, { Alpha } } } That can even be stated as: P = { , , } = = { Gamma, { Beta, { Alpha } } } = { Alpha <-> Beta <-> Gamma } where the symbol '<->' implies the successor/predecessor functions across a simple chain. To get full induction, and so infinity, we first extend the naturals to N* in a constructive framework, where we can count on both a lower and an upper bound. 0 = pred(1) 1 = succ(0) = pred(2) ... K = succ(K-1) = pred(K+1) ... N = succ(N-1) = pred(oo) oo = succ(N) Note that number 'N' above may be virtual (when we speak in theory, or if we are in a system that handles numbers of arbitrary size). Anyway, in N* hold the properties: -- forany k in N*: k != oo => exists succ(k) [prop.oo] And its converse: -- forany k in N*: k != 0 => exists pred(k) [prop.0] Now, we explicit the construction of the P set, to show a coformality between our simple chains and the naturals (extended naturals, for the case of infinite chains), and a simple way to state a bijection. Let: K-1 = pred(K) K+1 = succ(K) Given the birecursive closed chain: 0 = { } -- lower closure [prop.0] 1 = { 0 } 2 = { 1, { 0 } } 3 = { 2, { 1, { 0 } } } ... K+1 = { K, { K-1 } } ... oo = { N, { N-1 } } -- upper closure [prop.oo] We can see here that our first element can be put in correspondence to the empty set, and so on along the chain. Then we can see that our construction can be stated in terms of the successor/predecessor functions, so we can put our sets in bijection with natural numbers, again starting from the empty set. ord-Objects <-> ord-Sets* <-> Ordinals (*As defined, and I guess there may be different ways.) I hope it's not all rubbish. How does it sound? Is that what you where trying to avoid? I am a student in case you wandered... -LV -LV Bijectively mapped to a second set of equal cardinality doesn't seem > catchy. I wouldn't know... It seems to me the numbers above are still just > names. Anyway, P shows a structure that N lacks. BTW, in relational calculus we have relations and eventually their > indexes... -LV === Subject: Re: What's the good word? posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > If I have eight objects [...] [Sorry, corrected and final one.] P = { , , } = = { Gamma, { Beta, { Alpha } } } = { Alpha <-> Beta <-> Gamma } where the symbol '<->' implies the successor/predecessor functions across a simple chain. To get full induction, and so infinity, we first extend the naturals > to N* in a constructive framework, where we can count on both a lower > and an upper bound. 0 = pred(1) > 1 = succ(0) = pred(2) > ... > K = succ(K-1) = pred(K+1) > ... > N = succ(N-1) = pred(oo) > oo = succ(N) Note that number 'N' above may be virtual (when we speak in theory, or > if we are in a system that handles numbers of arbitrary size). Anyway, in N* hold the properties: -- forany k in N*: k != oo => exists succ(k) [prop.oo] And its converse: -- forany k in N*: k != 0 => exists pred(k) [prop.0] Now, we explicit the construction of the P set, to show a coformality > between our simple chains and the naturals (extended naturals, for the > case of infinite chains), and a simple way to state a bijection. Let: K-1 = pred(K) > K+1 = succ(K) Given the birecursive closed chain: 0 = { } -- lower closure [prop.0] > 1 = { 0 } > 2 = { 1, { 0 } } > 3 = { 2, { 1, { 0 } } } > ... > K+1 = { K, { K-1 } } > ... > oo = { N, { N-1 } } -- upper closure [prop.oo] We can see here that our first element can be put in correspondence to > the empty set, and so on along the chain. Then we can see that our > construction can be stated in terms of the successor/predecessor > functions, so we can put our sets in bijection with natural numbers, > again starting from the empty set. ord-Objects <-> ord-Sets* <-> Ordinals (*As defined, and I guess there may be different ways.) I hope it's not all rubbish. How does it sound? Is that what you where > trying to avoid? I am a student in case you wandered... Bijectively mapped to a second set of equal cardinality doesn't seem > catchy. I wouldn't know... It seems to me the numbers above are still just > names. Anyway, P shows a structure that N lacks. BTW, in relational calculus we have relations and eventually their > indexes... -LV === Subject: Re: What's the good word? posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > [...] Maybe there's not a name for every concept? Brilliant, thank you very much! -- the universe of facts (language) is in bijection with a subset of the universe of ideas (mind); past that, we risk to get philosophical; -- a diagram can tell more than thousand words; a pictogram embeds the power of analogy; an idea with no name is a point with a mass, a whole sub-universe; -- each set is a predicate and each predicate is a set; still, there is not a name for every concept! -LV === Subject: Re: What's the good word? > If I have eight objects in slots labeled 1,2,...,8, then I say the > objects are ordered. If I have the same eight objects inserted in slots labeled Tom, > Fred,....,Haggis,Horton, then I've imposed some order on them -- but > I have not implied any ordering relation as inherent in the > integers. What, if anything, is the shorthand for this idea? Bijectively mapped to a second set of equal cardinality doesn't seem > catchy. Nominal or categorical. Phil H === Subject: Re: What's the good word? days. My association with the Department is that of an alumnus. >If I have eight objects in slots labeled 1,2,...,8, then I say the >objects are ordered. If I have the same eight objects inserted in slots labeled Tom, >Fred,....,Haggis,Horton, then I've imposed some order on them -- but >I have not implied any ordering relation as inherent in the >integers. What, if anything, is the shorthand for this idea? Combinations refers to selections in which order is irrelevant. Permutations refers to selections in which order is important. -- Arturo Magidin magidin-at-member-ams-org === Subject: Re: What's the good word? posting-account=mV9EXQoAAACmCMM9qg0N4eJlXyr2Z93U CLR 1.1.4322),gzip(gfe),gzip(gfe) If I have eight objects in slots labeled 1,2,...,8, then I say the >objects are ordered. If I have the same eight objects inserted in slots labeled Tom, >Fred,....,Haggis,Horton, then I've imposed some order on them -- but >I have not implied any ordering relation as inherent in the >integers. æWhat, if anything, is the shorthand for this idea? Combinations refers to selections in which order is > irrelevant. Permutations refers to selections in which order is > important. I was asking what we call mappings or labelings which don't happen to be ordered. Maybe labeling. :-) > If I have eight objects in slots labeled 1,2,...,8, then I say the > objects are ordered. If I have the same eight objects inserted in slots labeled Tom, > Fred,....,Haggis,Horton, then I've imposed some order on them -- but > I have not implied any ordering relation as inherent in the > integers. What, if anything, is the shorthand for this idea? Bijectively mapped to a second set of equal cardinality doesn't seem > catchy. Nominal or categorical. Hmm... so instead of saying consider an ordered set, I ought to say consider a nominal set or consider a categorical set? Is that about the right idiom? How about a labeled or structured set -- the latter implying the mapping is onto some structure, if no necessarily ordering? It occurs to me that the general idea is function; an ordered set is one with a function into the integers. We could have a function into a name set, an index set, a set of properties, or anything at all. Or maybe you will say a partially ordered set may not admit of this description. It simply has a partial binary relation: a < b or a Hmm... so instead of saying consider an ordered set, I ought to say > consider a nominal set or consider a categorical set? Is that > about the right idiom? How about a labeled or structured set -- > the latter implying the mapping is onto some structure, if no > necessarily ordering? Maybe a catagorized set? Anyways, just define your term once at the start of your paper and press on... === Subject: Re: Matrix Algebra question posting-account=83nIfQoAAABGeBEFpUvAim95beEXyt31 1.1.4322; InfoPath.2; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) >Let L 2 be the 2x2 lower triangular matrix whose nonzero off diagonal >entry is 2, i.e. a11=1, a12=0, >a21=2, a22=1. æLet U 2 be its transpose. >I am looking for an easy proof of the following fact: The group (with matrix multiplication) generated by {L 2, U 2} is the >set of matrices A with a11, a22 odd, and a21, a12 even, and det(A)=1. A direct proof seems to be not easy. There's a simple direct proof in Nehari Conformal Mapping: > Say the group generated by those two matrices is G and the > group of all matrices such that a11 is odd[etc] is H. It's trivial to check that G is a subgroup of H. For the > other direction: Define chi([[a,b],[c,d]]) = |a| + |c|. > Say A is in H, and let S = {BA : B in G}. Say C is > an element of S that minimizes chi. You can show > by contradiction that C {2,1} = 0, and then it > follows that C is a power of U 2. If I have read this question correctly, then I don't think it is > correct. The group generated by L and U is actually free (it has index > 12 in the full modular group SL(2,Z)), and does not contain the > element -I (which has order 2), but -I is in H. By the induction argument on |a|+|c|, you can show show that, for > every g in H, either g or -g is in G. To be a bit more explicit, given any g = [[a,b], [c,d]] in æH, æby > multiplying g on the left by a generator of G or its inverse, you can > replace æa by a+2c or a-2c, or c by c+2a or c-2a. So unless b=0 one of > these options will always reduce the larger of |a| and |c|, > without changing the other, æthereby reducing |a|+|c|. So eventually you get a odd and b=0, but then determinant=1 forces a=1 > or -1, so c=a. Now multiplying by a power of L gets you to I or -I. Sorry, replace b=0 by c=0 and c=a by d=a in above. Derek Holt.- Hide quoted text - - Show quoted text - I think you are right. So Rudin wasn't careful in his book Real and Complex Analysis section 16.19. So what is the correct characterization of G then? -TCL === Subject: Re: Matrix Algebra question posting-account=-PngCgkAAAD2yUjosqWv1Nf1lkqWP4lp rv:1.8.1.14) Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) Let L_2 be the 2x2 lower triangular matrix whose nonzero off diagonal >entry is 2, i.e. a11=1, a12=0, >a21=2, a22=1. Let U_2 be its transpose. >I am looking for an easy proof of the following fact: The group (with matrix multiplication) generated by {L_2, U_2} is the >set of matrices A with a11, a22 odd, and a21, a12 even, and det(A)=1. A direct proof seems to be not easy. There's a simple direct proof in Nehari Conformal Mapping: > Say the group generated by those two matrices is G and the > group of all matrices such that a11 is odd[etc] is H. It's trivial to check that G is a subgroup of H. For the > other direction: Define chi([[a,b],[c,d]]) = |a| + |c|. > Say A is in H, and let S = {BA : B in G}. Say C is > an element of S that minimizes chi. You can show > by contradiction that C_{2,1} = 0, and then it > follows that C is a power of U_2. If I have read this question correctly, then I don't think it is > correct. The group generated by L and U is actually free (it has index > 12 in the full modular group SL(2,Z)), and does not contain the > element -I (which has order 2), but -I is in H. By the induction argument on |a|+|c|, you can show show that, for > every g in H, either g or -g is in G. To be a bit more explicit, given any g = [[a,b], [c,d]] in H, by > multiplying g on the left by a generator of G or its inverse, you can > replace a by a+2c or a-2c, or c by c+2a or c-2a. So unless b=0 one of > these options will always reduce the larger of |a| and |c|, > without changing the other, thereby reducing |a|+|c|. So eventually you get a odd and b=0, but then determinant=1 forces a=1 > or -1, so c=a. Now multiplying by a power of L gets you to I or -I. Sorry, replace b=0 by c=0 and c=a by d=a in above. Derek Holt.- Hide quoted text - - Show quoted text - I think you are right. So Rudin wasn't careful in his book Real and > Complex Analysis section 16.19. > So what is the correct characterization of G then? > -TCL In fact the diagonal entries of G are easily seen to be congruent to 1 mod 4, so that, together with off-diagonal entries even, characterizes G. Derek Holt. === Subject: Re: Matrix Algebra question posting-account=83nIfQoAAABGeBEFpUvAim95beEXyt31 1.1.4322; InfoPath.2; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) Let L 2 be the 2x2 lower triangular matrix whose nonzero off diagonal >entry is 2, i.e. a11=1, a12=0, >a21=2, a22=1. æLet U 2 be its transpose. >I am looking for an easy proof of the following fact: The group (with matrix multiplication) generated by {L 2, U 2} is the >set of matrices A with a11, a22 odd, and a21, a12 even, and det(A)=1. A direct proof seems to be not easy. There's a simple direct proof in Nehari Conformal Mapping: > Say the group generated by those two matrices is G and the > group of all matrices such that a11 is odd[etc] is H. It's trivial to check that G is a subgroup of H. For the > other direction: Define chi([[a,b],[c,d]]) = |a| + |c|. > Say A is in H, and let S = {BA : B in G}. Say C is > an element of S that minimizes chi. You can show > by contradiction that C {2,1} = 0, and then it > follows that C is a power of U 2. If I have read this question correctly, then I don't think it is > correct. The group generated by L and U is actually free (it has index > 12 in the full modular group SL(2,Z)), and does not contain the > element -I (which has order 2), but -I is in H. By the induction argument on |a|+|c|, you can show show that, for > every g in H, either g or -g is in G. To be a bit more explicit, given any g = [[a,b], [c,d]] in æH, æby > multiplying g on the left by a generator of G or its inverse, you can > replace æa by a+2c or a-2c, or c by c+2a or c-2a. So unless b=0 one of > these options will always reduce the larger of |a| and |c|, > without changing the other, æthereby reducing |a|+|c|. So eventually you get a odd and b=0, but then determinant=1 forces a=1 > or -1, so c=a. Now multiplying by a power of L gets you to I or -I. Sorry, replace b=0 by c=0 and c=a by d=a in above. Derek Holt.- Hide quoted text - - Show quoted text - I think you are right. So Rudin wasn't careful in his book Real and > Complex Analysis section 16.19. My mistake. Rudin was correct since A and -A correspond to same linear fractional transformation. -TCL > So what is the correct characterization of G then? > -TCL- Hide quoted text - - Show quoted text - === Subject: Re: Matrix Algebra question > Let L_2 be the 2x2 lower triangular matrix whose nonzero off diagonal > entry is 2, i.e. a11=1, a12=0, > a21=2, a22=1. Let U_2 be its transpose. > I am looking for an easy proof of the following fact: > The group (with matrix multiplication) generated by {L_2, U_2} is the > set of matrices A with a11, a22 odd, and a21, a12 even, and det(A)=1. > A direct proof seems to be not easy. > There's a simple direct proof in Nehari Conformal Mapping: > Say the group generated by those two matrices is G and the > group of all matrices such that a11 is odd[etc] is H. > It's trivial to check that G is a subgroup of H. For the > other direction: Define chi([[a,b],[c,d]]) = |a| + |c|. > Say A is in H, and let S = {BA : B in G}. Say C is > an element of S that minimizes chi. You can show > by contradiction that C_{2,1} = 0, and then it > follows that C is a power of U_2. > If I have read this question correctly, then I don't think it is > correct. The group generated by L and U is actually free (it has index > 12 in the full modular group SL(2,Z)), and does not contain the > element -I (which has order 2), but -I is in H. > By the induction argument on |a|+|c|, you can show show that, for > every g in H, either g or -g is in G. > To be a bit more explicit, given any g = [[a,b], [c,d]] in H, by > multiplying g on the left by a generator of G or its inverse, you can > replace a by a+2c or a-2c, or c by c+2a or c-2a. So unless b=0 one of > these options will always reduce the larger of |a| and |c|, > without changing the other, thereby reducing |a|+|c|. > So eventually you get a odd and b=0, but then determinant=1 forces a=1 > or -1, so c=a. Now multiplying by a power of L gets you to I or -I. > Sorry, replace b=0 by c=0 and c=a by d=a in above. > Derek Holt.- Hide quoted text - > - Show quoted text - > I think you are right. So Rudin wasn't careful in his book Real and > Complex Analysis section 16.19. My mistake. Rudin was correct since A and -A correspond to same linear > fractional transformation. > -TCL > So what is the correct characterization of G then? > -TCL- Hide quoted text - > - Show quoted text - G is the main congruence group Gamma[2] of matrices A in Sl(2,Z) with A = E mod 2. G is known to be generated by U, L and -E. This, e.g., drops out of a subtle comparison with the so-called theta group (cf. some preliminary chapter on the operation of subgroups of Sl(2,Z) on the upper half-plane in a book on elliptic modular forms). J. === Subject: What's the period of (cos ax)(sin bx) ? posting-account=hC93ZgoAAACbmsUh2kqn0V7FdT_S77NZ .NET CLR 3.5; ffco7) Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) Like the subject says. === Subject: Re: What's the period of (cos ax)(sin bx) ? sherifffruitfly a .8ecrit : > Like the subject says. > If m is the least common multiple of a and b (i.e. the smallest x such that x/a and x/b are integers), it is m/(2*pi). If there is no such m (like for a =1 and b=sqrt 2), the function is not periodic, but pseudo-periodic. === Subject: Re: What's the period of (cos ax)(sin bx) ? David C. Ullrich a .8ecrit : > sherifffruitfly a .8ecrit : > Like the subject says. > If m is the least common multiple of a and b (i.e. the smallest x such > that x/a and x/b are integers), it is m/(2*pi). If there is no such m > (like for a =1 and b=sqrt 2), the function is not periodic, but > pseudo-periodic. I think a more standard word (in English) for the concept I > think you're referring to would be almost periodic. Yes, I should have checked (it is presque p.8eriodique i.e. almost periodic in French too). Wikipedia gives (in English) a lot of important information on those, like the definition by completion of the set of sum of harmonic functions of arbitrary period. David C. Ullrich === Subject: Re: What's the period of (cos ax)(sin bx) ? If a and b are fractions of integers find a factor N such that N*a and N*b are integers. If a and b already are integers then N=1. let d = abs(a*N - b*N) if d is odd then Period = Pi*N if d is even then Period = 2*Pi*N if d =0 then Period = Pi/a If a or b are irrational numbers the period becomes infinity. Mark === Subject: Re: What's the period of (cos ax)(sin bx) ? <27276636.1213506473785.JavaMail.jakarta@nitrogen.mathforum.org>, > If a and b are fractions of integers find a factor N such > that N*a and N*b are integers. If a and b already are > integers then N=1. let d = abs(a*N - b*N) if d is odd then Period = Pi*N > if d is even then Period = 2*Pi*N > if d =0 then Period = Pi/a If a or b are irrational numbers the period becomes infinity. > Mark The function cos(a * x) * sin(b * x) will be periodic if and only if the ratio of a to b is rational. Otherwise it will be an almost-periodic function. http://en.wikipedia.org/wiki/Almost_periodic_function http://mathworld.wolfram.com/AlmostPeriodicFunction.html http://planetmath.org/encyclopedia/AlmostPeriodicFunction.html === Subject: Re: What's the period of (cos ax)(sin bx) ? Mark Hauerbach a .8ecrit : > If a and b are fractions of integers find a factor N such > that N*a and N*b are integers. If a and b already are > integers then N=1. let d = abs(a*N - b*N) if d is odd then Period = Pi*N > if d is even then Period = 2*Pi*N > if d =0 then Period = Pi/a If a or b are irrational numbers the period becomes infinity. > Mark Thre are a few istakes in this answzer. For instance, if a= pi, b=2 pi, a and b are irrationals, but the period is 2. === Subject: Help on constrained partitioning of integers posting-account=_srvNQoAAACo-MqlX20-xki6mTyxO02g Gecko/2008041514 Firefox/3.0b5,gzip(gfe),gzip(gfe) I'm working on a partition counting problem and i would require pointer to literature or a suggested method of approach on the same. Problem statement: Let r(N, b) be defined as the number of ways a positive integer N can be partitioned into positive integral parts such that the parts are all divisors of b and that the LCM of all the parts is equal to b. For example, r(8, 6) = 5 (excluding 8) 8 = 6 + 2 = 6 + 1 + 1 = 3 + 3 + 2 = 3 + 2 + 2 + 1 = 3 + 2 + 1 + 1 + 1 Ramasamy === Subject: Re: Help on constrained partitioning of integers posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY rv:1.8.1.12) Gecko/20080201 Firefox/2.0.0.12,gzip(gfe),gzip(gfe) > I'm working on a partition counting problem and i would require > pointer to literature or a suggested method of approach on the same. Problem statement: > Let r(N, b) be defined as the number of ways a positive integer N can > be partitioned into positive integral parts such that the parts are > all divisors of b and that the LCM of all the parts is equal to b. The generating function for partitions into parts dividing b is F_b(x) = product_{d|b}1/(1 - x^d). But you want the partitions into parts whose lcm is exactly b. These can be sieved out from those with parts dividing b. Let G_b(x) be the generating function. Then sum_{d|b} G_d(x) = F_d(x). By Mobius inversion, G_d(x) = sum_{d|b} mu(b/d) F_d(x). This can be simplified even more ... Victor Meldrew I don't believe it! === Subject: Re: Help on constrained partitioning of integers posting-account=_srvNQoAAACo-MqlX20-xki6mTyxO02g Gecko/2008060309 Firefox/3.0,gzip(gfe),gzip(gfe) I'm working on a partition counting problem and i would require > pointer to literature or a suggested method of approach on the same. Problem statement: > Let r(N, b) be defined as the number of ways a positive integer N can > be partitioned into positive integral parts such that the parts are > all divisors of b and that the LCM of all the parts is equal to b. The generating function for partitions into parts dividing b is > F b(x) = product {d|b}1/(1 - x^d). > But you want the partitions into parts whose lcm is exactly b. > These can be sieved out from those with parts dividing b. > Let G b(x) be the generating function. Then sum {d|b} G d(x) = F d(x). > By Mobius inversion, G d(x) = sum {d|b} mu(b/d) F d(x). > This can be simplified even more ... Victor Meldrew > I don't believe it! counting it where the selection of divisors and counting up became algorithmic. === Subject: Re: This Week's Finds in Mathematical Physics (Week 264) > 4) Sidney Coleman, lecture notes on quantum field theory, > http://www.damtp.cam.ac.uk/user/dt281/qft/col1.pdf > and > http://www.damtp.cam.ac.uk/user/dt281/qft/col2.pdf Someone should LaTeX them up! If they're freely available (as this comment suggests), I'll place them into the Federation Archive, with coverage (sizably) expanded from the original. But PDF, from XML/Word. I quit using LaTex 20 years ago. === Subject: Re: This Week's Finds in Mathematical Physics (Week 264) > 4) Sidney Coleman, lecture notes on quantum field theory, >http://www.damtp.cam.ac.uk/user/dt281/qft/col1.pdf > and >http://www.damtp.cam.ac.uk/user/dt281/qft/col2.pdf Someone should LaTeX them up! If they're freely available (as this comment suggests), I'll place > them into the Federation Archive, with coverage (sizably) expanded > from the original. But PDF, from XML/Word. I quit using LaTex 20 years > ago. I'm happy to hear that. I always had the feeling that LaTeX was something designed to keep out the unwashed public. PDF is much more democratic. no comment, for once. === Subject: Re: This Week's Finds in Mathematical Physics (Week 264) > 4) Sidney Coleman, lecture notes on quantum field theory, >http://www.damtp.cam.ac.uk/user/dt281/qft/col1.pdf > and >http://www.damtp.cam.ac.uk/user/dt281/qft/col2.pdf Someone should LaTeX them up! If they're freely available (as this comment suggests), I'll place > them into the Federation Archive, with coverage (sizably) expanded > from the original. But PDF, from XML/Word. I quit using LaTex 20 years > ago. P.S. What is the Federation Archive? === Subject: Re: Question about representations of Lie algebra sl(n). posting-account=UybuvQoAAACYu4uby_0_HMpWpMO1pzZl .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > Let V æbe irreducible finitedimension representation of Lie algebra > sl(n) æand let Lambda={lambda 1, lambda 2, .....} æ be the set of > all weights of the representation. The set Lambda is invariant under > the Weil W group æ action. How is possyble to describe in Lambda æthe > orbits æ æ of the action? æI mean - need effective description of the > sets W(lambda 1), W(lanbda 2),.. , æmaybe in terms æof roots. In each orbit W(lambda) there is a unique dominant weight w lying in the fundamental Weyl chamber. These dominant weights w can be efficiently obtained from the highest weight w0 by subtracting positive roots, keeping the weight dominant. The orbit W(w) bijectively corresponds to the group W if w is not orthogonal to any simple root, and to the group W 1 generated by reflections with respect to simle roots *not* orthogonal to w. So to describe the orbit effectively, you need an enumeration of the elements of the Weyl group for the root system and for any subset of its simple roots. === Subject: Re: Question about representations of Lie algebra sl(n). posting-account=jxLm8AoAAAD2bMDjnPVAuL2xlLkQFla8 8.54,gzip(gfe),gzip(gfe) Let V æbe irreducible finitedimension representation of Lie algebra > sl(n) æand let Lambda={lambda 1, lambda 2, .....} æ be the set of > all weights of the representation. The set Lambda is invariant under > the Weil W group æ action. How is possyble to describe in Lambda æthe > orbits æ æ of the action? æI mean - need effective description of the > sets W(lambda 1), W(lanbda 2),.. , æmaybe in terms æof roots. In each orbit W(lambda) there is a unique dominant weight w lying in > the fundamental Weyl chamber. Why unique? For example. let us consider representation with highest weight (3,3) of sl(2). Then orbits of the Weil group are concentic hexogens. It ie easy to see (sinse roots are (2,-1),(-1,2)) that the dominant weights (2,2),(0,3),(3,0) lies on the same orbit. What is my mistake? These dominant weights w can be > efficiently obtained from the highest weight w0 by subtracting > positive roots, keeping the weight dominant. The orbit W(w) > bijectively corresponds to the group W if w is not orthogonal to any > simple root, and to the group W 1 generated by reflections with > respect to simle roots *not* orthogonal to w. So to describe the orbit > effectively, you need an enumeration of the elements of the Weyl group > for the root system and for any subset of its simple roots. === Subject: Re: Maximum number of points closest to a point? === Subject: Relative primes. posting-account=WR65RgoAAACboMXdKKYDn4ZKOJhMgvsz Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) What is the easiest (and shortest) way to prove that for all natural numbers n there is m in {n,...,n+5} such that for all t in {n,...,n+5}/ {m} (t,m)=1? For what natural number x does it hold that for all natural numbers n there is m in {n,...,n+x} such that for all t in {n,...,n+5}/{m} (t,m)=1? === Subject: Re: Relative primes. posting-account=H1y7YgoAAADzGQwbcYaL9UvwttgsjOjp AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.1 Safari/525.18,gzip(gfe),gzip(gfe) > What is the easiest (and shortest) way to prove that for all natural > numbers n there is m in {n,...,n+5} such that for all t in {n,...,n+5}/ > {m} (t,m)=1? > For what natural number x does it hold that for all natural numbers n > there is m in {n,...,n+x} such that for all t in {n,...,n+5}/{m} > (t,m)=1? Didn't we just have this? Suppose n is odd. At least one of the numbers n + 2, n + 4 is not a multiple of 3; show that said number is prime to all the others. If n is even, work with n + 1 and n + 3. For the other part, I draw your attention to the numbers 2184, ..., 2200. GM === Subject: Re: Relative primes. posting-account=WR65RgoAAACboMXdKKYDn4ZKOJhMgvsz Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) What is the easiest (and shortest) way to prove that for all natural > numbers n there is m in {n,...,n+5} such that for all t in {n,...,n+5}/ > {m} (t,m)=1? > For what natural number x does it hold that for all natural numbers n > there is m in {n,...,n+x} such that for all t in {n,...,n+5}/{m} > (t,m)=1? Didn't we just have this? Well we did. I was only wondering can it be done in some other way because of my second question. > Suppose n is odd. At least one of the numbers n + 2, n + 4 > is not a multiple of 3; show that said number is prime to all > the others. If n is even, work with n + 1 and n + 3. For the other part, I draw your attention to the numbers > 2184, ..., 2200. (2184,2200)=8 (2185,2200)=5 (2186,2200)=2 (2187,2199)=3 (2188,2200)=4 (2189,2200)=11 (2190,2200)=10 (2191,2198)=7 (2192,2200)=8 (2193,2199)=3 (2194,2200)=2 (2195,2200)=5 (2196,2200)=4 (2197,2184)=13 (2198,2200)=2 (2199,2196)=3 (2200,2198)=2 Indeed there is not such element in {2184,...,2200} which is relative prime to all others. How did you get this example? This shows that {n,...,n+16} does not satisfy the condition however, it still does not answer the question for what x does {n,...,n+x} satisfy the condition. > __ > GM === Subject: Re: Die Antinomie unendlicher Anzahlen <4852baa3$0$4375$5402220f@news.sunrise.ch> posting-account=6xUtvgkAAAD_jypmLa2oo2HnrV0e8X9q > Albrecht æ> Der A-Automat E erzeugt die > æ> Folge der Quadratzahlen von 1 beginnend, > æ> indem er in jedem Schritt an zwei > æ> aneinanderstossende Seiten Punkte hinzuf.9fgt, also > æ æ> X > æ æ> XY > æ> YY > æ æ> XXY > æ> XXY > æ> YYY > æ æ> XXXY > æ> XXXY > æ> XXXY > æ> YYYY > æ æ> ... > æ æ> Die Y verdeutlichen die im jeweiligen > æ> Schritt hinzugef.9fgten Punkte. > æ æ> Ist das Resultat nun sowohl waagrecht > æ> wie auch senkrecht unendlich, Auf Grund der A4-Form des Papieres auf das > dies ausgedruckt wird und abh.8angig von der > verwendeten Schriftart, ist das Resuloltat > horizontal unendlicher als vertikal. Anderes Resultat: > Die x sind undenlicher als die Y :-] æ> oder nicht? Unendlich ist ein nicht und nirgends > existieredner Irrtum der Matehmatiker. æ> Was sagt die Mengenlehre dazu? Die sagt nix. Bloss deren Vertreter sagen da > - je nach Mengenlehre-Konfession - > etwas anderes. -- > ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ > mfg > Hans Josshttp://www.hjp.ch/ Ich kann durchaus akzeptieren, dass die unendliche Menge als eine facon de parler eine sinnvolle Anwendung in der Mathematik hat. Man darf dabei aber den besonderen Status dieses Hilfsmittels nicht vergessen, aehnlich vielleicht wie bei dem Umgang mit dem Differenzialquotienten, bei dem Umgang mit der Null in der Regel von d'Hospital, etc. Weniger bewusst aber genau so diffizil der Umgang mit Zahlen wie 0.1111 ... und ganz allgemein unendliche Summen. Letztlich geht es mir nicht um die Verwendung unendlicher Mengen oder Klassen in der Mathematik, sondern um so unsinnige Konsequenz wie die des Diagonalargumentes von Cantor, die genau auf diesen unkritischen Umgang mit angeblichen unendlichen Gesamtheiten basiert. Gruss Albrecht === Subject: Re: Die Antinomie unendlicher Anzahlen > Albrecht æ> Der A-Automat E erzeugt die > æ> Folge der Quadratzahlen von 1 beginnend, > æ> indem er in jedem Schritt an zwei > æ> aneinanderstossende Seiten Punkte hinzuf.9fgt, > also > æ æ> X > æ æ> XY > æ> YY > æ æ> XXY > æ> XXY > æ> YYY > æ æ> XXXY > æ> XXXY > æ> XXXY > æ> YYYY > æ æ> ... > æ æ> Die Y verdeutlichen die im jeweiligen > æ> Schritt hinzugef.9fgten Punkte. > æ æ> Ist das Resultat nun sowohl waagrecht > æ> wie auch senkrecht unendlich, Auf Grund der A4-Form des Papieres auf das > dies ausgedruckt wird und abh.8angig von der > verwendeten Schriftart, ist das Resuloltat > horizontal unendlicher als vertikal. Anderes Resultat: > Die x sind undenlicher als die Y :-] æ> oder nicht? Unendlich ist ein nicht und nirgends > existieredner Irrtum der Matehmatiker. æ> Was sagt die Mengenlehre dazu? Die sagt nix. Bloss deren Vertreter sagen da > - je nach Mengenlehre-Konfession - > etwas anderes. -- > ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ > mfg > Hans Josshttp://www.hjp.ch/ Ich kann durchaus akzeptieren, dass die unendliche > Menge als eine > facon de parler eine sinnvolle Anwendung in der > Mathematik hat. Man > darf dabei aber den besonderen Status dieses > Hilfsmittels nicht > vergessen, aehnlich vielleicht wie bei dem Umgang mit > dem > Differenzialquotienten, bei dem Umgang mit der Null > in der Regel von > d'Hospital, etc. > Weniger bewusst aber genau so diffizil der Umgang mit > Zahlen wie > 0.1111 ... und ganz allgemein unendliche Summen. Letztlich geht es mir nicht um die Verwendung > unendlicher Mengen oder > Klassen in der Mathematik, sondern um so unsinnige > Konsequenz wie die > des Diagonalargumentes von Cantor, die genau auf > diesen unkritischen > Umgang mit angeblichen unendlichen Gesamtheiten > basiert. Gruss > Albrecht Vielleicht dass wird wertfoll ins zu English uebersetzen ??? 0.1111... ist aber (is just) 1/9 und ich bin der Meinung, dass die alle andere Serien mit Ihren Automat geschrieben, leider sehr weit in Vergleichung zum Cantor oder Dedekind Theorien liegen. (I think, that all other series used to be written with Your automat are very far from Cantor or Dedekind theories indeed. ) Anyhow Your writing maschine needs so much time and so much instructions, that it is less practical then geometrical symbols and lines... (Dabei Ihre Schreibenmaschine verlangt so viel Zeit und so viele Instructionen, dass veniger praktisch als die Geometrische Figuren und Linien ist...) Verzeihung/Pardon Ro-Bin === Subject: Re: Die Antinomie unendlicher Anzahlen posting-account=6xUtvgkAAAD_jypmLa2oo2HnrV0e8X9q > Meine bisher beste Formulierung der Antinomie unendlicher Anzahlen: Ein absolut zuverl?aessiger, unzerst?oerbarer, ewig laufender Automat mit > unendlichem Ausgaberaum ist ein A-Automat (Aeternity-Automat oder > auch, ganz unbescheiden, Albrecht-Automat :-)). Gegeben sei ein A-Automat A der nichts weiter macht als eine Liste der > Form X > XX > XXX > XXXX > XXXXX > ... zu erzeugen. > Gegeben sei weiterhin ein A-Automat B der nichts weiter macht als eine > Zeile der Form XXXXXXXX ... zu schreiben. So wie die nat?uerlichen Zahlen nach Aufstellen der Peano-Axiome sofort > als aktual unendliche Menge vorliegen (im modernen Verst?aendnis der > Mathematik), so liegt auch das Ergebnis des A-Automaten nach dessen > Definition fest und kann als gegeben und vorliegend angenommen werden. Der A-Automat A erzeugt eine Liste nat?uerlicher Zahlen in unitae?rer > Schreibweise. Jede dieser Zahlen ist endlich. Folglich enthae?lt die > Ausgabe des A-Automaten A in jeder Zeile nur endlich viele Zeichen. > Die erste Spalte enth?aelt aber unendlich viele Zeichen, da es > unendlich viele nat?uerliche Zahlen gibt. Wie verh?aelt es sich nun mit dem A-Automaten B? Da der A-Automat B > unendlich oft Zeichen um Zeichen in die Zeile schreibt, muss diese > Zeile unendlich viele Zeichen enthalten. Nun betrachten wir einen A-Automat C. Dieser Automat schreibt im > ersten Zyklus ein ?X? in eine Zeile. In jedem weiteren Zyklus tut er > folgendes: > Er dupliziert die vorhandene Zeile in eine zweite Zeile und verl?aengert > diese Zeile mit einem ?X?. Er erzeugt also den Nachfolger. Dann l?oescht > er die erstere Zeile, so dass nur noch die Zeile mit dem Nachfolger > vorhanden ist. > Dieser Zyklus wird unendlich oft wiederholt. Wie sieht nun die Ausgabe des A-Automaten C aus? Enth?aelt die Ausgabe > eine Zeile mit einem unendlichen String wie A-Automat B? Oder ist der > String in der Zeile jederzeit nur endlich wie beim A-Automat A? Damit > w?aere die Ausgabe aber undefiniert, da es keine gr?oesste nat?uerliche Zahl > gibt. Nun macht der A-Automat C aber nichts anderes wie der A-Automat B, nur > das er zwischen jeder Verl?aengerung des Strings weitere Arbeitschritte > ausf?uehrt, die aber f?uer das Endergebnis unerheblich sind. > Der A-Automat C macht aber auch nichts anderes wie der A-Automat A, > nur mit dem Unterschied, dass er die jeweilige Vorg?ngerzeile l?oescht. Damit muss der A-Automat C aber gleichzeitig einen String definierter > Laenge, n?aemlich der Lae?nge ?Unendlich? und einen undefinierten String, > n?aemlich ?gr?oesste nat?uerliche Zahl? erzeugen. Eine offensichtliche Antinomie. Auch nett: Ein weiterer A-Automat D hat folgendes Programm: Er > verl?aengert in jedem Schritt sowohl den String in der ersten Zeile wie > auch den in der ersten Spalte. Die Ausgabe sae?he also so aus: XXXXXXXXXXXXX ? > X > X > X > X > X > . > . > . Sind beide Strings unendlich oder nicht? K?oennen sie verschieden sein? > Warum muss dann beim A-Automat A nicht auch ein unendlicher String in > einer Zeile stehen? Hier eine nette Variation: Der A-Automat E erzeugt die Folge der Quadratzahlen von 1 beginnend, > indem er in jedem Schritt an zwei aneinanderstossende Seiten Punkte > hinzufuegt, also X XY > YY XXY > XXY > YYY XXXY > XXXY > XXXY > YYYY ... Die Y verdeutlichen die im jeweiligen Schritt hinzugefuegten Punkte. Ist das Resultat nun sowohl waagrecht wie auch senkrecht unendlich, > oder nicht? Oder gar senkrecht unendlich und waagrecht unbestimmt > endlich??? Oder anders herum? Oder gar nicht definiert??? Was sagt die Mengenlehre dazu? > Gruss > Albrecht S. Storz Noch ein A-Automat: Der A-Automat F wird mit folgender Anweisung beschrieben: Zuerst schreibt er in die erste Zeile ein Zeichen X. In jedem weiteren Schritt erzeugt der Automat vor den vorhandenen Zeilen eine neue Zeile und schreibt darin ein Zeichen mehr als in der darauffolgenden Zeile Zeichen stehen. Der Output wird also so entwickelt: X XX X XXX XX X XXXX XXX XX X ... Wie sieht die erste Zeile der vollstaendigen (unendlichen) Entwicklung aus? === Subject: Re: String Theory Summarized <61e52$484984f5$12753@news.teranews.com> posting-account=jPnQ2goAAAA461y3QD0lbyw0oKeThma1 AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.1 Safari/525.20,gzip(gfe),gzip(gfe) what I really wanted to know, was the comparison of using compressed natural gas (methane) to liquid H2 that is made from it, but first not bothering with losses for handling; just the stoichiometries. as for superstrings, I'd guess that they are completely supereseded by superbranes, especially if most of the string & loop formalism derives from tearing the pants of the phase-spaces involved ... well, that's hard to visualize if a string is 1d, especially if the 1d is considered to be time; did they acutally compactify the time, or just one or more arbitrary spatial dims? > in bulk, . Now consider, forgetting all of MeOH's undesirable > æqualities, that MeOH is essentially a CH fuel that contains > With only 44% of Methanol's weight being burnable a 100% > amount (comparable to Gasoline's caloric content) would thus: I should extirpate monsieur Klein's name from Kaluza's, since he is the progenitor of the original compactified dimensional scheme.... well, he also gave us the first funnystring! so, like, was there ever any mention of which dimension was compactified, or is it just/otherwise the same as the other three ... time is NOT a dimension, or it is the only dimension of any consequence (yay, Bucky .-) thus quoth: he really does show Smolin's DOUBLEplusUNwrongs. http://cosmicvariance.com/2006/12/07/guest-blogger-joe-polchinski-on-... http://xkcd.com/171/ http://arxiv.org/PS cache/arxiv/pdf/0802/0802.3249v1.pdf -UN HYDROGEN (sic; Methanex (TM) reformanteurs) ECONOMIE?... La Troi Phases d'Exploitation de la Protocols des Grises de Kyoto: (FOSSILISATION [McCainanites] (TM/sic))/ BORE/GUSH/NADIR @ http://www.tarpley.net/aobook.htm. Http://www.tarpley.net/bushb.htm (content partiale, below): 17 -- L'ATTEMPTER de COUP D'ETAT, 3/30/81 === Subject: Re: String Theory Summarized <61e52$484984f5$12753@news.teranews.com> posting-account=jPnQ2goAAAA461y3QD0lbyw0oKeThma1 AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.1 Safari/525.20,gzip(gfe),gzip(gfe) still haven't found that quotage. anyway, I should extirpate monsieur Klein's name from Kaluza's, since he is the progenitor of the original compactified dimensional scheme.... well, he also gave us hte first funnystring! so, like, was there ever any mention of which dimension was compactified, or is it just/otherwise the same as the other three ... time is NOT a dimension, or it is the only dimension of any consequence (yay, Bucky .-) > I will try to find the argument that I quoted, > before, about the heretofore resultage. the most tepid argumentarium is against the number > of dimensions, when Kaluza-Klein is so seemingly OK thus quoth: he really does show Smolin's doubleplusunwrongs. http://cosmicvariance.com/2006/12/07/guest-blogger-joe-polchinski-on-... -UN HYDROGEN (sic; Methanex (TM) reformanteurs) ECONOMIE?... La Troi Phases d'Exploitation de la Protocols des Grises de Kyoto: (FOSSILISATION [McCainanites] (TM/sic))/ BORE/GUSH/NADIR @ http://www.tarpley.net/aobook.htm. Http://www.tarpley.net/bushb.htm (content partiale, below): 17 -- L'ATTEMPTER de COUP D'ETAT, 3/30/81 === Subject: Re: Should-be-simple algebra question posting-account=4W9t0woAAAB3dq7-fhllNhT3_mnzEKiC Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) On each of the five sides of a regular pentagon there are three > circles - two on each corner and one at the mid-point FOR writing in > whole numbers (there should be 10 circles). The top side middle circle > is 21, top-right side middle circle is 32, bottom-right side middle > circle is 47, bottom-left side middle circle is 56 and top-left side > middle circle is 10. The sum of each side's three numbers is 100. Fill > in the corner numbers. I tried doing it with many simultaneous equations and substitution but > I found myself always with two variables, one on each side of the > equals sign. I'm never able to find out one of the values. The ten numbers, in order from the downward point going clockwise are: > A , 56 , B , 10 , C , 21 , D , 32 , E , 47 Let A = x, then by looking ate each side consecutively, we have: > B = 44 - x, C = 46 + x, D = 33 - x, E = 35 + x, A = 18 - x > Thus x = 18 - x, and so A = 9, b = 35, C = 55, D = 24, and E = 44. The second part of the question says to show there's only way of doing this. Does this already show that there's only one way since you've expressed all unknown values in terms of one variable x and found from solving these equations that x equals only one value? === Subject: Re: Should-be-simple algebra question > On each of the five sides of a regular pentagon there are three > circles - two on each corner and one at the mid-point FOR writing in > whole numbers (there should be 10 circles). The top side middle circle > is 21, top-right side middle circle is 32, bottom-right side middle > circle is 47, bottom-left side middle circle is 56 and top-left side > middle circle is 10. The sum of each side's three numbers is 100. Fill > in the corner numbers. The second part of the question says to show there's only way of doing > this. Five linear equation, five unknowns, there is a general theorem that one can appeal to. -- === Subject: Re: Should-be-simple algebra question posting-account=ogMREwkAAAC5xUr8sg7heGtsvzzF18LA Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) On Jun 15, 9:25 am, Frederick Williams On each of the five sides of a regular pentagon there are three > circles - two on each corner and one at the mid-point FOR writing in > whole numbers (there should be 10 circles). The top side middle circle > is 21, top-right side middle circle is 32, bottom-right side middle > circle is 47, bottom-left side middle circle is 56 and top-left side > middle circle is 10. The sum of each side's three numbers is 100. Fill > in the corner numbers. The second part of the question says to show there's only way of doing > this. Yes, since the last equation (which I solved for x) has only one solution, there is only one answer. Note that it is NOT because I expressed everything in terms one one variable (although since all equations are linear, the two are closely related- the final equation is an equation in one variable, and so will have 0, 1, or infinite solutions. 0 if you end up with something like x = 20 + x, and infinite if you end up with x = x. Since this is a pentagon, those two situations cannot arise (since the RHS will have the opposite sign on first, however) > Five linear equation, five unknowns, there is a general theorem that one > can appeal to. This is not true, unless you do some extra work first. Any system of equations can be inconsistant, or have multiple solutions. For linear equations, having the number of equations at least equal to the number of variables is a necessary, but not sufficient, condition to having a unique solution. To show you can have more, note that given any system with a unique solution, you can add more (redundant) equations. To show that equal numbers of equations and unknowns is not necessarily enough, note the following systems of 3 equations and 3 unknowns: 1) x + y = 1 x + + z = 2 2x + y + z = 3 2) x + y = 1 x + + z = 2 2x + y + z = 4 The first has infinite solutions, the second has no solution. === Subject: Re: Comprehensive Solution Manual for Textbooks posting-account=gPBcvAoAAABZFDMYmWX7vWDRGgmyzC6G InfoPath.1),gzip(gfe),gzip(gfe) Federal Taxation 2007: Comprehensive - Thomas Pope (20th ed) (ISBN 0132389479). === Subject: Re: Problem of differential calculus > Please help me solve the following problem of IIT examination: > f(x) is defined for x>-1 and has a continuous derivative. It satisfies > f(0)=1; f'(0) = 0 and (1+f(x)) f(x) = 1+x, then, solve the > following three problems: > (a) Graph of f'(x) is: > (i) Always Convex (ii) Always concave (iii) first concave then convex > (iv) first convex then concave > (b) Max value of f'(1) is: > (i) 1 (ii) 1/3 (iii) 2/3 (iv) 3/4 > (c) f'(-1) is always: > (i) >= +1/4 (ii) <= 1/4 (iii) >= -1/4 (iv) < = -1/4 f is defined for x > -1, so what does f'(-1) mean? Actually, the function is well-defined at x = -1, so that is the least of my worries. But what does it mean to say that f'(-1) is _always_ <= -1/4. I mean, either it is or it isn't. Also (c) i-iv are not exclusive. -- Timothy Murphy e-mail: gayleard /at/ eircom.net tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Problem of differential calculus > Please help me solve the following problem of IIT examination: > f(x) is defined for x>-1 and has a continuous derivative. It satisfies > f(0)=1; f'(0) = 0 and (1+f(x)) f(x) = 1+x, then, solve the > following three problems: > (a) Graph of f'(x) is: > (i) Always Convex (ii) Always concave (iii) first concave then convex > (iv) first convex then concave > (b) Max value of f'(1) is: > (i) 1 (ii) 1/3 (iii) 2/3 (iv) 3/4 > (c) f'(-1) is always: > (i) >= +1/4 (ii) <= 1/4 (iii) >= -1/4 (iv) < = -1/4 f is defined for x > -1, so what does f'(-1) mean? Actually, the function is well-defined at x = -1, > so that is the least of my worries. You need to read the hypotheses again. > But what does it mean to say that f'(-1) is _always_ <= -1/4. > I mean, either it is or it isn't. > Also (c) i-iv are not exclusive. === Subject: Re: Problem of differential calculus > Please help me solve the following problem of IIT examination: > f(x) is defined for x>-1 and has a continuous derivative. It satisfies > f(0)=1; f'(0) = 0 and (1+f(x)) f(x) = 1+x, then, solve the > following three problems: (a) Graph of f'(x) is: > (i) Always Convex (ii) Always concave (iii) first concave then convex > (iv) first convex then concave (b) Max value of f'(1) is: > (i) 1 (ii) 1/3 (iii) 2/3 (iv) 3/4 (c) f'(-1) is always: > (i) >= +1/4 (ii) <= 1/4 (iii) >= -1/4 (iv) < = -1/4 > f is defined for x > -1, so what does f'(-1) mean? > Actually, the function is well-defined at x = -1, > so that is the least of my worries. You need to read the hypotheses again. I realise that the OP stated that his function was defined for x > -1. However, the differential equation he gives defines a function on a range (c,infty) where c < -1. Therefore it makes sense to talk of f'(-1). Nb I agree that the OP was very careless in copying down what must have been a precise question, since it apparently appeared in an exam. I think it is quite an interesting problem. I'd like to know how people approach it. -- Timothy Murphy e-mail: gayleard /at/ eircom.net tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Calculus Question posting-account=el6zfgoAAAB2wn47zpNOWEYlvViuQXyN 1.1.4322),gzip(gfe),gzip(gfe) Question: It tells me to integrate this function by Integration by Parts: (Integral) e^(6x) sin e^(3x) dx I know how to Integrate by Parts, however, when I completed the problem and checked the answer, my answer was way off! I looked at the way they solved it, and I found a slight difference in how the Integral was solved. I set: u= e^(6x); dv= sin e^(3x) dx as I should first, and set the rest accordingly. However, they set: u=e^(3x); dv=e^(3x) sin e^(3x) dx This, I do believe, is how the variation between our answers came about. However, I don't understand why they divided e^(3x) between u and dv. Could you help me to better understand why? -S.L._Calc === Subject: Re: Calculus Question posting-account=cTANYAoAAAADW4iDglUZ0T-yFcwscdzW Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) Question: It tells me to integrate this function by Integration by Parts: (Integral) e^(6x) sin e^(3x) dx I know how to Integrate by Parts, however, when I completed the > problem and checked the answer, my answer was way off! I looked at the > way they solved it, and I found a slight difference in how the > Integral was solved. I set: u= e^(6x); dv= sin e^(3x) dx as I should first, and set the rest > accordingly. However, they set: u=e^(3x); dv=e^(3x) sin e^(3x) dx This, I do believe, is how the variation between our answers came > about. However, I don't understand why they divided e^(3x) between u and dv. Could you help me to better understand why? > -S.L._Calc I've worked it out two different ways; first by tables and second by parts. Parts is by far the longer method. product rule: diff [uv] = u * diff {v] + v * diff [u] int by parts: int v du = uv - int u dv set v = sin exp 3x; du = exp 6x dx therefore, dv = cos exp 3x * 3exp3x dx; u = 6^-1 * exp 6x and, int v du = 6^-1 * exp6x * sin exp3x - int 6^-1 * exp6x * cos exp3x * 3exp3x dx repeat a 2nd time until cos under int becomes neg sin and combine like terms from original equation. Woila! *** Or, just do this: let y = exp6x therefore, dy = 6exp6x then the sub looks like: 6^-1 * int dy * sin y^1/2 use tables for this integral and substitute back into x. === Subject: Re: Calculus Question Question: It tells me to integrate this function by Integration by Parts: (Integral) e^(6x) sin e^(3x) dx I know how to Integrate by Parts, however, when I completed the >problem and checked the answer, my answer was way off! I looked at the >way they solved it, and I found a slight difference in how the >Integral was solved. I set: u= e^(6x); dv= sin e^(3x) dx as I should first, and set the rest >accordingly. However, they set: u=e^(3x); dv=e^(3x) sin e^(3x) dx This, I do believe, is how the variation between our answers came >about. However, I don't understand why they divided e^(3x) between u and dv. Could you help me to better understand why? By noticing that e^(6x) = e^(3x) * e^(3x), the integral can be written Int e^(3x) sin[e^(3x)] e^(3x) dx If one now makes the substitution t = e^(3x), then dt = 3 e^(3x) dx, and the integral becomes 1/3 Int t sin(t) dt which is now easy to integrate by parts. >-S.L. Calc === Subject: Re: Calculus Question posting-account=el6zfgoAAAB2wn47zpNOWEYlvViuQXyN 1.1.4322),gzip(gfe),gzip(gfe) æQuestion: It tells me to integrate this function by Integration by Parts: (Integral) e^(6x) sin e^(3x) dx I know how to Integrate by Parts, however, when I completed the > problem and checked the answer, my answer was way off! I looked at the > way they solved it, and I found a slight difference in how the > Integral was solved. I set: u= e^(6x); dv= sin e^(3x) dx as I should first, and set the rest > accordingly. However, they set: u=e^(3x); dv=e^(3x) sin e^(3x) dx This, I do believe, is how the variation between our answers came > about. However, I don't understand why they divided e^(3x) between u and dv. Could you help me to better understand why? > -S.L. Calc Doesn't Anyone Know Calculus? Please Help Me! I can't see my help source for another 3 days! Inquiringly, -S.L. Calc === Subject: Re: Calculus Question > æQuestion: > It tells me to integrate this function by Integration by Parts: > (Integral) e^(6x) sin e^(3x) dx > I know how to Integrate by Parts, however, when I completed the > problem and checked the answer, my answer was way off! I looked at the > way they solved it, and I found a slight difference in how the > Integral was solved. > I set: > u= e^(6x); dv= sin e^(3x) dx as I should first, and set the rest > accordingly. > However, they set: > u=e^(3x); dv=e^(3x) sin e^(3x) dx > This, I do believe, is how the variation between our answers came > about. > However, I don't understand why they divided e^(3x) between u and dv. > Could you help me to better understand why? > -S.L._Calc Doesn't Anyone Know Calculus? >Please Help Me! >I can't see my help source for another 3 days! What are we, a 24-hour help line? You only waited one hour before posting your reply. Also keep in mind it's Saturday. (Also note that questions that look like they're homework tend to go unanswered - and you did mention school at one point.) Usually, the answer to why it was done a certain way is, it's just something you learn to do - in this case, dv can be rewritten as dv = 1/3 (sin e^(3x)) d(e^(3x)) so (if you substitute z for e^3x) you get u dv = z sin z dz (the integral of which is z cos z - sin z, or e^3x cos e^3x - sin e^3x) which is much more manageable than setting u = e^6x and dv = sin e^3x dx (to be honest, I would have done the same thing). -- Don === Subject: Re: Calculus Question <08u85495a178qo0v6b72pljvon1419fmfc@4ax.com> posting-account=el6zfgoAAAB2wn47zpNOWEYlvViuQXyN 1.1.4322),gzip(gfe),gzip(gfe) On Jun 14, 9:14æpm, Don Del Grande æQuestion: > It tells me to integrate this function by Integration by Parts: > (Integral) e^(6x) sin e^(3x) dx > I know how to Integrate by Parts, however, when I completed the > problem and checked the answer, my answer was way off! I looked at the > way they solved it, and I found a slight difference in how the > Integral was solved. > I set: > u= e^(6x); dv= sin e^(3x) dx as I should first, and set the rest > accordingly. > However, they set: > u=e^(3x); dv=e^(3x) sin e^(3x) dx > This, I do believe, is how the variation between our answers came > about. > However, I don't understand why they divided e^(3x) between u and dv. > Could you help me to better understand why? > -S.L. Calc Doesn't Anyone Know Calculus? >Please Help Me! >I can't see my help source for another 3 days! What are we, a 24-hour help line? æYou only waited one hour before > posting your reply. æAlso keep in mind it's Saturday. (Also note that questions that look like they're homework tend to go > unanswered - and you did mention school at one point.) Usually, the answer to why it was done a certain way is, it's just > something you learn to do - in this case, dv can be rewritten as > dv = 1/3 (sin e^(3x)) d(e^(3x)) > so (if you substitute z for e^3x) you get > u dv = z sin z dz > (the integral of which is z cos z - sin z, or > e^3x cos e^3x - sin e^3x) > which is much more manageable than setting u = e^6x and dv = sin e^3x > dx (to be honest, I would have done the same thing). -- Don- Hide quoted text - - Show quoted text - I'm sorry Don. I came off as entitled. It kind of gets to my head when I do 4 kinds of Advanced Math every day. I do appriciate everyone's help, and just so we're clear on this, that problem was no more than an extra practice problem I did to hone my skills in the area. S.L. Calc === Subject: Re: Calculus Question <08u85495a178qo0v6b72pljvon1419fmfc@4ax.com> posting-account=el6zfgoAAAB2wn47zpNOWEYlvViuQXyN 1.1.4322),gzip(gfe),gzip(gfe) On Jun 14, 9:14æpm, Don Del Grande æQuestion: > It tells me to integrate this function by Integration by Parts: > (Integral) e^(6x) sin e^(3x) dx > I know how to Integrate by Parts, however, when I completed the > problem and checked the answer, my answer was way off! I looked at the > way they solved it, and I found a slight difference in how the > Integral was solved. > I set: > u= e^(6x); dv= sin e^(3x) dx as I should first, and set the rest > accordingly. > However, they set: > u=e^(3x); dv=e^(3x) sin e^(3x) dx > This, I do believe, is how the variation between our answers came > about. > However, I don't understand why they divided e^(3x) between u and dv. > Could you help me to better understand why? > -S.L. Calc Doesn't Anyone Know Calculus? >Please Help Me! >I can't see my help source for another 3 days! What are we, a 24-hour help line? æYou only waited one hour before > posting your reply. æAlso keep in mind it's Saturday. (Also note that questions that look like they're homework tend to go > unanswered - and you did mention school at one point.) Usually, the answer to why it was done a certain way is, it's just > something you learn to do - in this case, dv can be rewritten as > dv = 1/3 (sin e^(3x)) d(e^(3x)) > so (if you substitute z for e^3x) you get > u dv = z sin z dz > (the integral of which is z cos z - sin z, or > e^3x cos e^3x - sin e^3x) > which is much more manageable than setting u = e^6x and dv = sin e^3x > dx (to be honest, I would have done the same thing). -- Don- Hide quoted text - - Show quoted text - I'm sorry Don. When you do Advanced Mathematics all day long, like I do, I kind of gets to your head. I acted in an entitled fashion and I apologize. S.L. Calc === Subject: Re: Calculus Question <2fp954th1roedasssoi6ac5s3fb9ek8c70@4ax.com> posting-account=el6zfgoAAAB2wn47zpNOWEYlvViuQXyN 1.1.4322),gzip(gfe),gzip(gfe) On Jun 14, 9:14æpm, Don Del Grande problem and checked the answer, my answer was way off! I looked at the > way they solved it, and I found a slight difference in how the > Integral was solved. I set: u= e^(6x); dv= sin e^(3x) dx as I should first, and set the rest > accordingly. However, they set: u=e^(3x); dv=e^(3x) sin e^(3x) dx This, I do believe, is how the variation between our answers came > about. However, I don't understand why they divided e^(3x) between u and dv. Could you help me to better understand why? > -S.L. Calc >Doesn't Anyone Know Calculus? >Please Help Me! >I can't see my help source for another 3 days! > What are we, a 24-hour help line? æYou only waited one hour before > posting your reply. æAlso keep in mind it's Saturday. > (Also note that questions that look like they're homework tend to go > unanswered - and you did mention school at one point.) > Usually, the answer to why it was done a certain way is, it's just > something you learn to do - in this case, dv can be rewritten as > dv = 1/3 (sin e^(3x)) d(e^(3x)) > so (if you substitute z for e^3x) you get > u dv = z sin z dz > (the integral of which is z cos z - sin z, or > e^3x cos e^3x - sin e^3x) > which is much more manageable than setting u = e^6x and dv = sin e^3x > dx (to be honest, I would have done the same thing). > -- Don- Hide quoted text - > - Show quoted text - I'm sorry Don. When you do Advanced Mathematics all day long, like I >do, I kind of gets to your head. Seems very strange that someone who does Advanced Mathematics > all day would still have trouble with Elementary Mathematics > like first-year calculus. I acted in an entitled fashion and I apologize. S.L. Calc David C. Ullrich- Hide quoted text - - Show quoted text - Mr. Ullrich... Hmmm... are you from MIT, perhaps? === Subject: Re: Calculus Question posting-account=el6zfgoAAAB2wn47zpNOWEYlvViuQXyN 1.1.4322),gzip(gfe),gzip(gfe) >On Jun 14, 9:14æpm, Don Del Grande problem and checked the answer, my answer was way off! I looked at the > way they solved it, and I found a slight difference in how the > Integral was solved. I set: u= e^(6x); dv= sin e^(3x) dx as I should first, and set the rest > accordingly. However, they set: u=e^(3x); dv=e^(3x) sin e^(3x) dx This, I do believe, is how the variation between our answers came > about. However, I don't understand why they divided e^(3x) between u and dv. Could you help me to better understand why? > -S.L. Calc >Doesn't Anyone Know Calculus? >Please Help Me! >I can't see my help source for another 3 days! > What are we, a 24-hour help line? æYou only waited one hour before > posting your reply. æAlso keep in mind it's Saturday. > (Also note that questions that look like they're homework tend to go > unanswered - and you did mention school at one point.) > Usually, the answer to why it was done a certain way is, it's just > something you learn to do - in this case, dv can be rewritten as > dv = 1/3 (sin e^(3x)) d(e^(3x)) > so (if you substitute z for e^3x) you get > u dv = z sin z dz > (the integral of which is z cos z - sin z, or > e^3x cos e^3x - sin e^3x) > which is much more manageable than setting u = e^6x and dv = sin e^3x > dx (to be honest, I would have done the same thing). > -- Don- Hide quoted text - > - Show quoted text - I'm sorry Don. When you do Advanced Mathematics all day long, like I >do, I kind of gets to your head. Seems very strange that someone who does Advanced Mathematics > all day would still have trouble with Elementary Mathematics > like first-year calculus. I acted in an entitled fashion and I apologize. S.L. Calc David C. Ullrich- Hide quoted text - - Show quoted text - Mr. Ullrich... > Hmmm... > are you from MIT, perhaps?- Hide quoted text - - Show quoted text - I appear to be a strange type, don't I? I am. I..... Hmmmmm......................... === Subject: Re: Calculus Question <2fp954th1roedasssoi6ac5s3fb9ek8c70@4ax.com> posting-account=el6zfgoAAAB2wn47zpNOWEYlvViuQXyN 1.1.4322),gzip(gfe),gzip(gfe) On Jun 14, 9:14æpm, Don Del Grande problem and checked the answer, my answer was way off! I looked at the > way they solved it, and I found a slight difference in how the > Integral was solved. I set: u= e^(6x); dv= sin e^(3x) dx as I should first, and set the rest > accordingly. However, they set: u=e^(3x); dv=e^(3x) sin e^(3x) dx This, I do believe, is how the variation between our answers came > about. However, I don't understand why they divided e^(3x) between u and dv. Could you help me to better understand why? > -S.L. Calc >Doesn't Anyone Know Calculus? >Please Help Me! >I can't see my help source for another 3 days! > What are we, a 24-hour help line? æYou only waited one hour before > posting your reply. æAlso keep in mind it's Saturday. > (Also note that questions that look like they're homework tend to go > unanswered - and you did mention school at one point.) > Usually, the answer to why it was done a certain way is, it's just > something you learn to do - in this case, dv can be rewritten as > dv = 1/3 (sin e^(3x)) d(e^(3x)) > so (if you substitute z for e^3x) you get > u dv = z sin z dz > (the integral of which is z cos z - sin z, or > e^3x cos e^3x - sin e^3x) > which is much more manageable than setting u = e^6x and dv = sin e^3x > dx (to be honest, I would have done the same thing). > -- Don- Hide quoted text - > - Show quoted text - I'm sorry Don. When you do Advanced Mathematics all day long, like I >do, I kind of gets to your head. Seems very strange that someone who does Advanced Mathematics > all day would still have trouble with Elementary Mathematics > like first-year calculus. I acted in an entitled fashion and I apologize. S.L. Calc David C. Ullrich- Hide quoted text - - Show quoted text - Well, to be entirely honest, that little thing last night was just a test of how well you guys would respond to a friend in need. You all passed. That was part of a statistical analysis. -S.L. Calc === Subject: Re: Calculus Question posting-account=el6zfgoAAAB2wn47zpNOWEYlvViuQXyN 1.1.4322),gzip(gfe),gzip(gfe) >On Jun 14, 9:14æpm, Don Del Grande problem and checked the answer, my answer was way off! I looked at the > way they solved it, and I found a slight difference in how the > Integral was solved. I set: u= e^(6x); dv= sin e^(3x) dx as I should first, and set the rest > accordingly. However, they set: u=e^(3x); dv=e^(3x) sin e^(3x) dx This, I do believe, is how the variation between our answers came > about. However, I don't understand why they divided e^(3x) between u and dv. Could you help me to better understand why? > -S.L. Calc >Doesn't Anyone Know Calculus? >Please Help Me! >I can't see my help source for another 3 days! > What are we, a 24-hour help line? æYou only waited one hour before > posting your reply. æAlso keep in mind it's Saturday. > (Also note that questions that look like they're homework tend to go > unanswered - and you did mention school at one point.) > Usually, the answer to why it was done a certain way is, it's just > something you learn to do - in this case, dv can be rewritten as > dv = 1/3 (sin e^(3x)) d(e^(3x)) > so (if you substitute z for e^3x) you get > u dv = z sin z dz > (the integral of which is z cos z - sin z, or > e^3x cos e^3x - sin e^3x) > which is much more manageable than setting u = e^6x and dv = sin e^3x > dx (to be honest, I would have done the same thing). > -- Don- Hide quoted text - > - Show quoted text - I'm sorry Don. When you do Advanced Mathematics all day long, like I >do, I kind of gets to your head. Seems very strange that someone who does Advanced Mathematics > all day would still have trouble with Elementary Mathematics > like first-year calculus. I acted in an entitled fashion and I apologize. S.L. Calc David C. Ullrich- Hide quoted text - - Show quoted text - Well, to be entirely honest, that little thing last night was just a > test of how well you guys would respond to a friend in need. You all > passed. That was part of a statistical analysis. -S.L. Calc- Hide quoted text - - Show quoted text - I see... Hmmmmmm... I hope some one sees the latter post... -S.L. Calc === Subject: Re: Calculus Question posting-account=el6zfgoAAAB2wn47zpNOWEYlvViuQXyN 1.1.4322),gzip(gfe),gzip(gfe) > On Jun 14, 9:14æpm, Don Del Grande æQuestion: > It tells me to integrate this function by Integration by Parts: > (Integral) e^(6x) sin e^(3x) dx > I know how to Integrate by Parts, however, when I completed the > problem and checked the answer, my answer was way off! I looked at the > way they solved it, and I found a slight difference in how the > Integral was solved. > I set: > u= e^(6x); dv= sin e^(3x) dx as I should first, and set the rest > accordingly. > However, they set: > u=e^(3x); dv=e^(3x) sin e^(3x) dx > This, I do believe, is how the variation between our answers came > about. > However, I don't understand why they divided e^(3x) between u and dv. > Could you help me to better understand why? > -S.L. Calc Doesn't Anyone Know Calculus? >Please Help Me! >I can't see my help source for another 3 days! What are we, a 24-hour help line? æYou only waited one hour before > posting your reply. æAlso keep in mind it's Saturday. (Also note that questions that look like they're homework tend to go > unanswered - and you did mention school at one point.) Usually, the answer to why it was done a certain way is, it's just > something you learn to do - in this case, dv can be rewritten as > dv = 1/3 (sin e^(3x)) d(e^(3x)) > so (if you substitute z for e^3x) you get > u dv = z sin z dz > (the integral of which is z cos z - sin z, or > e^3x cos e^3x - sin e^3x) > which is much more manageable than setting u = e^6x and dv = sin e^3x > dx (to be honest, I would have done the same thing). -- Don- Hide quoted text - - Show quoted text - I'm sorry Don. When you do Advanced Mathematics all day long, like I > do, it kind of gets to your head. > I acted in an entitled fashion and I apologize. S.L. Calc- Hide quoted text - - Show quoted text - === Subject: Re: Calculus Question posting-account=el6zfgoAAAB2wn47zpNOWEYlvViuQXyN 1.1.4322),gzip(gfe),gzip(gfe) > æQuestion: It tells me to integrate this function by Integration by Parts: (Integral) e^(6x) sin e^(3x) dx I know how to Integrate by Parts, however, when I completed the > problem and checked the answer, my answer was way off! I looked at the > way they solved it, and I found a slight difference in how the > Integral was solved. I set: u= e^(6x); dv= sin e^(3x) dx as I should first, and set the rest > accordingly. However, they set: u=e^(3x); dv=e^(3x) sin e^(3x) dx This, I do believe, is how the variation between our answers came > about. However, I don't understand why they divided e^(3x) between u and dv. Could you help me to better understand why? > -S.L. Calc Doesn't Anyone Know Calculus? > Please Help Me! > I can't see my help source for another 3 days! > Inquiringly, > -S.L. Calc- Hide quoted text - - Show quoted text - I Guess No one is here tonight? *Please prove the latter wrong* Irritably, -S.L. Calc === Subject: Re: Calculus Question posting-account=el6zfgoAAAB2wn47zpNOWEYlvViuQXyN 1.1.4322),gzip(gfe),gzip(gfe) æQuestion: It tells me to integrate this function by Integration by Parts: (Integral) e^(6x) sin e^(3x) dx I know how to Integrate by Parts, however, when I completed the > problem and checked the answer, my answer was way off! I looked at the > way they solved it, and I found a slight difference in how the > Integral was solved. I set: u= e^(6x); dv= sin e^(3x) dx as I should first, and set the rest > accordingly. However, they set: u=e^(3x); dv=e^(3x) sin e^(3x) dx This, I do believe, is how the variation between our answers came > about. However, I don't understand why they divided e^(3x) between u and dv. Could you help me to better understand why? > -S.L. Calc Doesn't Anyone Know Calculus? > Please Help Me! > I can't see my help source for another 3 days! > Inquiringly, > -S.L. Calc- Hide quoted text - - Show quoted text - I Guess No one is here tonight? > *Please prove the latter wrong* Irritably, > -S.L. Calc- Hide quoted text - - Show quoted text - Alright, Joke's on Me! -Sighs- I get this at school, too... You guys are starting to irk me... Bye......... -Frowns Disappointedly- ..., -S.L. Calc === Subject: Re: Calculus Question posting-account=el6zfgoAAAB2wn47zpNOWEYlvViuQXyN 1.1.4322),gzip(gfe),gzip(gfe) > æQuestion: It tells me to integrate this function by Integration by Parts: (Integral) e^(6x) sin e^(3x) dx I know how to Integrate by Parts, however, when I completed the > problem and checked the answer, my answer was way off! I looked at the > way they solved it, and I found a slight difference in how the > Integral was solved. I set: u= e^(6x); dv= sin e^(3x) dx as I should first, and set the rest > accordingly. However, they set: u=e^(3x); dv=e^(3x) sin e^(3x) dx This, I do believe, is how the variation between our answers came > about. However, I don't understand why they divided e^(3x) between u and dv. Could you help me to better understand why? > -S.L. Calc Doesn't Anyone Know Calculus? > Please Help Me! > I can't see my help source for another 3 days! > Inquiringly, > -S.L. Calc- Hide quoted text - - Show quoted text - I Guess No one is here tonight? > *Please prove the latter wrong* Irritably, > -S.L. Calc- Hide quoted text - - Show quoted text - Alright, Joke's on Me! > -Sighs- I get this at school, too... > You guys are starting to irk me... > Bye......... > -Frowns Disappointedly- ..., > -S.L. Calc- Hide quoted text - - Show quoted text - ............................................................................ ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ...........................................................................w a iting....................................................................... . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ....................................- sighs-...................................................................... ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ................................................................... ............................................................................ ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ............................................................................ . ... ALRIGHT!!!!! I GIVE UP!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ANGRILY, S.L. Calc === Subject: Re: Will this theory fly? posting-account=8rQMIgoAAABFHTgT7kcJJwMeqT0fndEF 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) Once I am published, I'd love to debate these ideas with you anytime, anywhere. I assume you included quoatation marks because you assume (wrongly) > that in 2008 any ground-breaking theory must have been invented > already, or perhaps is a restating of something old. I made this > original post to see if anyone was able to take a wild shot how my > theory works, and no one came close... I certainly believe in ground-breaking ideas! æI also believe that most good > ideas are the result > of some experience. However, I think your explanation for your original post > is silly. > It reminds me of the game, I'm thinking of a number, I bet that you can't > guess it. Why not explain a bit of your theory, and see if it can hold up to the > scrutiny of this forum? > You haven't, so far, cleared any sort of hurdle that I can see. æMove > forward, as you put it... > Please don't be offended, but your reply sounds a little like it was written > by JSH. > Demonstrate for me (please!) that I am wrong. So, as a result of this little > experiment, I think I'm in the clear so far, and I can move forward. > Wouldn't you agree?- Hide quoted text - - Show quoted text - === Subject: Re: Will this theory fly? Once I am published, I'd love to debate these ideas with you anytime, anywhere. Have you published anything before? You seem unusually paranoid. I'm really skeptical about people with big ideas that don't need to familiarize themselves with what has already been done on a subject. You may as well say you've come up with a new proof to Fermat's Last Theorem.... You may as well say you are JSH... === Subject: Re: Will this theory fly? posting-account=8rQMIgoAAABFHTgT7kcJJwMeqT0fndEF 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) To be honest, to be VERY honest, I didn't want to just TELL my theory for fear of giving it away to someone who was probably a bystander - someone that likes to take other people's ideas and run with them, rather than professionally respond to the idea with counter-proposals or arguments. So I appologize if I treated you badly for fear of handing my ideas over to some unknown and unnamed person that may lurk out there. Also, this topic will be the subject of a future book being written in conjunction with a Java programmer. I wouldn't want to just give away my ideas, when I can eventually sell my ideas for $12.95 a pop.... I'm not stupid! lol I assume you included quoatation marks because you assume (wrongly) > that in 2008 any ground-breaking theory must have been invented > already, or perhaps is a restating of something old. I made this > original post to see if anyone was able to take a wild shot how my > theory works, and no one came close... I certainly believe in ground-breaking ideas! æI also believe that most good > ideas are the result > of some experience. However, I think your explanation for your original post > is silly. > It reminds me of the game, I'm thinking of a number, I bet that you can't > guess it. Why not explain a bit of your theory, and see if it can hold up to the > scrutiny of this forum? > You haven't, so far, cleared any sort of hurdle that I can see. æMove > forward, as you put it... > Please don't be offended, but your reply sounds a little like it was written > by JSH. > Demonstrate for me (please!) that I am wrong. So, as a result of this little > experiment, I think I'm in the clear so far, and I can move forward. > Wouldn't you agree?- Hide quoted text - - Show quoted text - === Subject: Re: Will this theory fly? >To be honest, to be VERY honest, I didn't want to just TELL my theory >for fear of giving it away to someone who was probably a bystander - >someone that likes to take other people's ideas and run with them, >rather than professionally respond to the idea with counter-proposals >or arguments. So I appologize if I treated you badly for fear of >handing my ideas over to some unknown and unnamed person that may lurk >out there. Also, this topic will be the subject of a future book being written in >conjunction with a Java programmer. I wouldn't want to just give away >my ideas, when I can eventually sell my ideas for $12.95 a pop.... I'm >not stupid! lol Why did I see this coming? You're just one of those cranks who has a brilliant revolutionary method (despite knowing nearly nothing of the field and the current the state of the art), which cannot be made public until it's copyrighted, patented, and NDA'd so that you will make billions of dollars (in your mind). The reality is you have nothing, and nobody cares. Go away. === Subject: Re: Will this theory fly? posting-account=8rQMIgoAAABFHTgT7kcJJwMeqT0fndEF 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) Obviously you aren't a terrorist. If a terrorist got a hold of my ideas, we might all be in trouble. On Jun 14, 7:36æam, Frederick Williams [...] Based on the amount of sunlight, the air temperature, and the number > of guards... when is the best time to attack? Comments?? I have entered building on many occasions. æI just walk in through the > doorway at the time that suits me. æI have never attacked. -- === Subject: Integrator posting-account=el6zfgoAAAB2wn47zpNOWEYlvViuQXyN 1.1.4322),gzip(gfe),gzip(gfe) Wolfram has done it again! http://integrals.wolfram.com/index.jsp === Subject: Measure theory - how to prove f is continuous? I'd like some help with this proof, please. Let A be a Lebesgue measurable subset of R and let A + x = {y in R | y = a + x, a in A} be the translation of A by the real number x. Show that the function defined on R by f(x) = m(A inter (A + x)) is continuous (m is the Lebesgue measure). I've been trying the following approach: 1)Represent A as a countable and increasing union, from inside, of compact and nice (hence measurable) sets, like intervals; 2)If A_n is the sequence that approximates A from inside, then define f_n(x) = f(A_n inter (A_n + x))so that each f_n is continuous and f is the uniform limit of f_n. This would prove the statement, but I think I choose a somewhat unfortunate and complicated approach. Artur === Subject: Re: Measure theory - how to prove f is continuous? posting-account=8wyvFgoAAAAJYWyfLHRzREe3lxFCHRTd MathPlayer 2.10b; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) > I'd like some help with this proof, please. Let A be a Lebesgue measurable subset of R and let A + x = {y in R | y = a + x, a in A} æbe the translation of A by the real number x. Show that the function defined on R by f(x) = m(A inter (A + x)) is continuous (m is the Lebesgue measure). I've been trying the following approach: 1)Represent A as a countable and increasing union, from inside, of æcompact and nice (hence measurable) sets, like intervals; 2)If A n is the sequence that approximates A from inside, then define f n(x) = f(A n inter (A n + x))so that each f n is continuous and f is the uniform limit of f n. This would prove the statement, but I think I choose a somewhat unfortunate and complicated approach. Artur Try writing f(x) = the integral of the product of the indicator (characteristic ) function g of A (g is 1 on A and 0 off A) and and the indicator function of A +x (namely h(y) =g(y-x) ) . Assume A has finite measure and use the bounded convergence theorem. If A has infinite measure then f has infinity as a possible value and I am not sure if f has to be continuous .For example A = the union of (n+ [0,1/10]) ,then it seems that f has a jump at x=1/10 from === Subject: Re: Measure theory - how to prove f is continuous? <2pp954p6b9gfku27tm9pmj9an5cejjq2db@4ax.com> posting-account=8wyvFgoAAAAJYWyfLHRzREe3lxFCHRTd MathPlayer 2.10b; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) > I'd like some help with this proof, please. > Let A be a Lebesgue measurable subset of R and let A + x = {y in R | y = a + x, a in A} æbe the translation of A by the real number x. Show that the function defined on R by f(x) = m(A inter (A + x)) is continuous (m is the Lebesgue measure). > I've been trying the following approach: 1)Represent A as a countable and increasing union, from inside, of æcompact and nice (hence measurable) sets, like intervals; 2)If A n is the sequence that approximates A from inside, then define f n(x) = f(A n inter (A n + x))so that each f n is continuous and f is the uniform limit of f n. > This would prove the statement, but I think I choose a somewhat unfortunate and complicated approach. > Artur Try writing f(x) = the integral of the product of the indicator >(characteristic ) function g æof A (g is 1 on A and 0 off A) and and >the indicator function of A +x (namely h(y) =g(y-x) ) . > æ æAssume A has finite measure and use the bounded convergence >theorem. I'm pretty sure that doesn't work. Say g is the indicator function > of A, so æ æ æ æf(x) = int g(y) g(y-x) dy, great. Assume A has finite measure, great (that was surely a > hypothesis that the OP just omitted). Now say x n -> x. > Exacly how do you show that g(y) g(y - x n) -> g(y) g(y - x) > for almost every y? (You don't, unless it happens that g is continuous almost > everywhere, which is not going to be true except for very > nice sets A.) Right but not that nice. If the boundary of A has measure 0 its ok .I wonder if the result holds otherwise. >If A has infinite measure then f has infinity as a possible >value and I am not sure if f has to be continuous .For example A = the >union of (n+ [0,1/10]) ,then it seems that f has a jump at x=1/10 from David C. Ullrich- Hide quoted text - - Show quoted text - === Subject: Problem of differential equations posting-account=hH_9UQoAAAAG4Vzyx-4ZVDXIWA0UKGHA 98),gzip(gfe),gzip(gfe) Please help me solve the following problem: f(x) is defined for x>-1 and has a continuous derivative. It satisfies f(0)=1; f'(0) = 0 and (1+f(x)) f(x) = 1+x, then, solve the following two problems: (a) Graph of f'(x) is: (i) Always Convex (ii) Always concave (iii) first concave then convex (iv) first convex then concave (b) Maximum value of f'(x) at x=1 is: (i) 1 (ii) 1/3 (iii) 2/3 (iv) 3/4 === Subject: Re: Problem of differential equations > Please help me solve the following problem: > f(x) is defined for x>-1 and has a continuous derivative. It > satisfies > f(0)=1; f'(0) = 0 and (1+f(x)) f(x) = 1+x, then, solve the > following two problems: (a) Graph of f'(x) is: > (i) Always Convex (ii) Always concave (iii) first concave then convex > (iv) first convex then concave It's fairly easy to see that f(x) > 0 for x >= 0, so f''(x) > 0. [If not, let a > 0 be the smallest solution of f(a) = 0. Then f''(x) > 0 on (0,a). Hence f'(x) > 0 on (0,a). Hence f(a) > f(0) on (0,a).] -- Timothy Murphy e-mail: gayleard /at/ eircom.net tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: Problem of differential equations > Please help me solve the following problem: > f(x) is defined for x>-1 and has a continuous derivative. It > satisfies > f(0)=1; f'(0) = 0 and (1+f(x)) f(x) = 1+x, then, solve the > following two problems: (a) Graph of f'(x) is: > (i) Always Convex (ii) Always concave (iii) first concave then convex > (iv) first convex then concave (b) Maximum value of f'(x) at x=1 is: > (i) 1 (ii) 1/3 (iii) 2/3 (iv) 3/4 You already asked this question, and I pointed out that (b) does not make sense as it stands. The function f(x) is well-defined for x > -1, so how can you speak of the _maximum_ value of f'(1)? You said before that the question came from an exam. I suggest you copy it more carefully. -- Timothy Murphy e-mail: gayleard /at/ eircom.net tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland === Subject: Re: JSH: Most Recent Cycle > They come, they try, they give up. It is a tribute to > mathematicians that there are people willing to go the extra mile > for the sake of the truth. It's a tribute to the cheap thrill of feeling superior to an ignorant > but loud person. > I don't think this applies to several people who I've seen expending considerable effort on trying to explain things to JSH. Their patience has been astonishing, and the word cheap certainly doesn't apply. === Subject: Re: JSH: Most Recent Cycle > They come, they try, they give up. It is a tribute to > mathematicians that there are people willing to go the extra mile > for the sake of the truth. > It's a tribute to the cheap thrill of feeling superior to an ignorant > but loud person. > I don't think this applies to several people who I've seen expending > considerable effort on trying to explain things to JSH. Their patience > has been astonishing, and the word cheap certainly doesn't apply. Sure, I was a bit churlish and there are a couple of exceptions, but by and large, I'd wager that JSH threads are kept alive by the cheap thrill I mention. Of course, I'm no stranger to that thrill myself. -- ...[W]hatever gifts I have, they are mine. And I do fully intend on NOT doing more research, NOT teaching, and NOT doing any number of things that other people may feel they have a right to tell me I should do, as when you had the chance with me, you crapped out. --James S. Harris === Subject: Re: JSH: Most Recent Cycle > They come, they try, they give up. It is a tribute to > mathematicians that there are people willing to go the extra mile > for the sake of the truth. It's a tribute to the cheap thrill of feeling superior to an ignorant > but loud person. > I don't think this applies to several people who I've seen expending > considerable effort on trying to explain things to JSH. Their patience > has been astonishing, and the word cheap certainly doesn't apply. Sure, I was a bit churlish and there are a couple of exceptions, but > by and large, I'd wager that JSH threads are kept alive by the cheap > thrill I mention. Oh, I suppose there's another motivation. Lots of us can't stand it when a braggart falsely claims a great accomplishment and concludes that he is superior to us common folk. This itch keeps a number of posters scratching, for no real good reason (JSH is fooling no one and doing no harm). Now, maybe you think that pointing out the false statements of a lying (or deluded) blowhard is for the sake of the truth, but I tend to think of the act in less noble terms. Not that it's a bad thing per se, but not a particularly noble calling either. > Of course, I'm no stranger to that thrill myself. -- And I'll reinforce the point that you are an enemy of humanity, that my predecessors are people like Gauss, Euler, Newton, Archimedes and others who you are spitting upon as you do it to me by trying to keep their discipline trashed as it is now. -- James S. Harris === Subject: Re: JSH: Most Recent Cycle > There are known knowns. There are things we know that we know. There > are known unknowns. That is to say there are things that we now know > we don't know. But there are also unknown unknowns. There are things > we don't know we don't know. -- Donald Rumsfield, Epistemologist are we missing unknown knowns? Obviously, Rumsfield is taking a stance: Epistemic logics must satisfy the rule: Kp -> KKp (If I know that p, then I know that I know that p.) Thus, Rumsfield is committed to S4 for epistemic logic, at least if he is also committed to the following two conditions: K(p -> q) -> (Kp -> Kq) (If I know that p implies q and I know that p, then I know that q.) Kp -> p (If I know that p, then p.) See http://plato.stanford.edu/entries/logic-epistemic/, though the author is surprisingly reticent on Rumsfield's contributions to the field. Must be a Democrat. -- Sure, [my Usenet presence is] like Shaq playing against you in your backyard, but that has its perks, as I find ways to have my fun *and* I can send messages to certain people in the United States Government without concern that the rest of you understand them. -- James Harris === Subject: Re: JSH: Most Recent Cycle > They come, they try, they give up. It is a tribute to > mathematicians that there are people willing to go the extra mile > for the sake of the truth. > It's a tribute to the cheap thrill of feeling superior to an ignorant > but loud person. You say that like it's a bad thing. Surely not! -- Jesse F. Hughes A year ago, my approval rating was in the 30s, my nominee for the Supreme Court had just withdrawn, and my vice president had shot someone. Ah, those were the good old days. G. W. Bush 3/28/07 === Subject: Re: simple nonabelian groups Crossed product asked in http://mathforum.org/kb/thread.jspa?messageID=6254373 > If A and B are simple nonabelian group, what are the > normal subgroups of A x B ? 1 x 1, 1 x B, A x 1, and A x B. If H is normal in A x B, and (a,b) in H and (c,d) in A x B, then [ (a,b), (c,d) ] = ( [a,c], [b,d] ) is in H. In particular, taking b=d, ( [a,c], 1 ) is in H. Hence [ pi1(H), A ] x 1 is a subgroup of H, but [ pi1(H), A ] is a normal subgroup of A, so either: * pi1(H) = 1 and H <= 1 x B, or * [ pi1(H), A ] = A and 1 x A <= H. Switching the roles of A and B gives another dichotomy, and the four cases are exactly the subgroups I listed. === Subject: Re: subgroups of finite index Crossedproduct asked in http://mathforum.org/kb/thread.jspa?messageID=6254365 > If H < G has finite index in G, > is it always true that [G : N_G(H)] =< [G : H] Yes. [G:H] = [G:N_G(H)]*[N_G(H):H]. > Also, if H has finite index in G, why does G > necessarily contain a proper normal subgroup of > finite index? This is a FAQ, check the archive of even the last month. G acts by multiplication on the finitely many cosets of H, hence there is a homomorphism from G into the symmetric group on [G:H] points, and so the kernel of that homomorphism is a normal subgroup of index at most [G:H]!. === Subject: Re: Primes of the form 2^(2^n)+1 <2uN4k.17729$_03.6044@reader1.news.saunalahti.fi> posting-account=fVOpuAkAAAB0gOUkQMH0DG_KdwTVgKXP CLR 2.0.50727),gzip(gfe),gzip(gfe) On Jun 14, 6:57æpm, Aatu Koskensilta Also I said (and I don't think anyone disagreed) that logically > simple means can be concisely expressed in ZFC. Concisely in what sense? Pretty much no statement of ordinary > mathematics can be concisely expressed in ZFC in the usual sense of > the word. Should you doubt this, just try to express, say, the > fundamental theorem of arithmetic in the language of set theory. I don't think you paid sufficient attention to my use of the past tense -- Also I said... This should have clued you in to the possibility that previous postings of mine on this thread might have answered your question. My previous postings gave my definition of concise as being relative to other unsolved problems of mathematics and I cited Thurston's Geometrization Conjecture as an example for comparison. I would think that the ZFC statement that the number of Fermat primes is finite would be shorter than the ZFC statement of Thurston's Geometrization Conjecture. Paul Epstein === Subject: Integral Extensions and Prime ideals posting-account=4m6i3AoAAACkBWCtXNbpbnHMq5T0a0Fu I know or believe to know that for any two commutative unitary semiprime rings A and B with A a subring of B, if A is integrally closed in B then it is not necessary true that A / p / A is integrally closed in B/ p for any prime ideal p in Spec B and vice versa .. that if A is a subring of B and A/ p / A is integrally closed in B/p for any prime ideal p in Spec B then A is not necessarily integrally closed in B. I was trying to think of examples, maybe some kind soul can help me out here =) Jose Capco === Subject: Analysis with integral, lebesgue... Hello teacher~ lim{n->oo} [int{0 to 1} f_n(x) dx] = int{0 to 1} [lim{n->oo} f_n(x)] dx Choose the sample that above equation does not hold. (1) f_n(x) = nx{(1 - x^2)^n} (2) f_n(x) = (x^n) / n (3) f_n(x) = x^n (4) f_n(x) = {sin(nx^2)} / n (5) f_n(x) = (nx) / {1 + (n^2.x^2)} Answer : (1) ------------------------------------------------------- (2), (3), (4), (5) <= 1 (bounded) for all n, x in [0,1]. so, by Bounded convergench theorem OR Lebesgue dominated covergence theorem, (2), (3), (4), (5) satisfy above equation. so, (1) is the anwer intuitively. How do you show that above equation does not hold on (1) ? Namely, how do you show that lim{n->oo} [int{0 to 1} {nx{(1 - x^2)^n}} dx] > 0. Of course, int{0 to 1} [lim{n->oo} {nx{(1 - x^2)^n}} dx = 0. === Subject: Re: Analysis with integral, lebesgue... > Hello teacher~ lim{n->oo} [int{0 to 1} f_n(x) dx] = int{0 to 1} [lim{n->oo} f_n(x)] dx Choose the sample that above equation does not hold. (1) f_n(x) = nx{(1 - x^2)^n} > (2) f_n(x) = (x^n) / n > (3) f_n(x) = x^n > (4) f_n(x) = {sin(nx^2)} / n > (5) f_n(x) = (nx) / {1 + (n^2.x^2)} Answer : (1) > ------------------------------------------------------- > (2), (3), (4), (5) <= 1 (bounded) for all n, x in [0,1]. > so, by Bounded convergench theorem OR > Lebesgue dominated covergence theorem, > (2), (3), (4), (5) satisfy above equation. so, (1) is the anwer intuitively. How do you show that above equation does not hold on (1) ? > Namely, how do you show that > lim{n->oo} [int{0 to 1} {nx{(1 - x^2)^n}} dx] > 0. Partial integration? > Of course, > int{0 to 1} [lim{n->oo} {nx{(1 - x^2)^n}} dx = 0. -- Best wishes, J. === Subject: Re: Analysis with integral, lebesgue... > Hello teacher~ > lim{n->oo} [int{0 to 1} f_n(x) dx] = int{0 to 1} [lim{n->oo} f_n(x)] dx > Choose the sample that above equation does not hold. > (1) f_n(x) = nx{(1 - x^2)^n} > (2) f_n(x) = (x^n) / n > (3) f_n(x) = x^n > (4) f_n(x) = {sin(nx^2)} / n > (5) f_n(x) = (nx) / {1 + (n^2.x^2)} > Answer : (1) > ------------------------------------------------------- > (2), (3), (4), (5) <= 1 (bounded) for all n, x in [0,1]. > so, by Bounded convergench theorem OR > Lebesgue dominated covergence theorem, > (2), (3), (4), (5) satisfy above equation. > so, (1) is the anwer intuitively. > How do you show that above equation does not hold on (1) ? > Namely, how do you show that > lim{n->oo} [int{0 to 1} {nx{(1 - x^2)^n}} dx] > 0. Partial integration? Sorry. easy by indefinite integral. int{0 to 1} {nx{(1 - x^2)^n}} dx = [n*{1/(n+1)}*{(1-x^2)^(n+1)}*(-1/2)]_{0 to 1} = n / (2n+2) ---> 1/2 as n ->oo. === Subject: Calculus with double integral.. Hello teacher~ R : the interior and boundary of the triangle with (0, 0), (-1/2, 0), (0, 1). Double int{R} sin{(y - 2x)^2} dA = ? Answer : (1/4)(1 - cos1) -------------------------------------------- Sorry, I need your advice. === Subject: Re: Calculus with double integral.. posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > Hello teacher~ R : the interior and boundary of the triangle with (0, 0), (-1/2, 0), (0, > 1). Double int{R} sin{(y - 2x)^2} dA = ? Answer : (1/4)(1 - cos1) -------------------------------------------- > Sorry, I need your advice. Express the integral as a repeated integral of the form int_{x=a..b}[int_{y=L(x)..U(x) f(x,y) dy] dx. By drawing a diagram, figure out what are a, b, L(x) and U(x). Then do the inner integral, followed by the outer integral. That would be my advice. R.G. Vickson === Subject: Re: Calculus with double integral.. > Hello teacher~ > R : the interior and boundary of the triangle with (0, 0), (-1/2, 0), (0, > 1). > Double int{R} sin{(y - 2x)^2} dA = ? > Answer : (1/4)(1 - cos1) > -------------------------------------------- > Sorry, I need your advice. Express the integral as a repeated integral of the form > int_{x=a..b}[int_{y=L(x)..U(x) f(x,y) dy] dx. By drawing a diagram, > figure out what are a, b, L(x) and U(x). Then do the inner integral, > followed by the outer integral. That would be my advice. Hm, int{0 to -1/2} [int{0 to 2x+1} sin{(y - 2x)^2} dy] dx. int{0 to 2x+1} sin{(y - 2x)^2} dy] , How do you calculate it ? === Subject: Re: Calculus with double integral.. > R : the interior and boundary of the triangle with (0, 0), (-1/2, 0), (0, > 1). > Double int{R} sin{(y - 2x)^2} dA = ? > Answer : (1/4)(1 - cos1) > -------------------------------------------- > Sorry, I need your advice. Express the integral as a repeated integral of the form > int_{x=a..b}[int_{y=L(x)..U(x) f(x,y) dy] dx. By drawing a diagram, > figure out what are a, b, L(x) and U(x). Then do the inner integral, > followed by the outer integral. That would be my advice. Did you follow your own advice? I tried to follow that approach. The problem (at least for me) is the do the inner integral part. Jose Carlos Santos === Subject: Re: Calculus with double integral.. <6bks4eF3cb5iiU1@mid.individual.net> posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > R : the interior and boundary of the triangle with (0, 0), (-1/2, 0), (0, > 1). > Double int{R} sin{(y - 2x)^2} dA = ? > Answer : (1/4)(1 - cos1) > -------------------------------------------- > Sorry, I need your advice. Express the integral as a repeated integral of the form > int {x=a..b}[int {y=L(x)..U(x) f(x,y) dy] dx. By drawing a diagram, > figure out what are a, b, L(x) and U(x). Then do the inner integral, > followed by the outer integral. That would be my advice. Did you follow your own advice? No, I did not. Mea Culpa. I suppose another way would be to perform a change of variables to u = (y - 2x)^2, v = something, and try to get a sensible 'v'. However, that might only transfer the nastiness from the inner integral to the outer integral. R.G. Vickson > I tried to follow that approach. The > problem (at least for me) is the do the inner integral part. > Jose Carlos Santos === Subject: Re: Calculus with double integral.. posting-account=F63B7goAAABsKhI4GVDJRhjUnH-QokKQ CLR 2.0.50727),gzip(gfe),gzip(gfe) Small modification, put: u = y - 2x v = -x Calculate Jacobian, change the integration area and evaluate new double integral. Vincent Vega --- http://lemmath.com ::::: Math problems with complete solutions === Subject: Re: Calculus with double integral.. Small modification, put: u = y - 2x > v = -x Calculate Jacobian, change the integration area and evaluate new > double integral. Yes, good idea. so, |dx/dy dx/dv| |dy/du dy/dv| = |(-1/2) -1| | 1 0| = 1 so, by tranformation of coordinate, int{0 to 1} [int{0 to (1/2)u} sin(u^2) dv] du = int{0 to 1} [(1/2).u.{sin(u^2)}] du = (1/4).(1 - cos1) === Subject: Re: Calculus with double integral.. > R : the interior and boundary of the triangle with (0, 0), (-1/2, 0), > (0, 1). > Double int{R} sin{(y - 2x)^2} dA = ? > Answer : (1/4)(1 - cos1) > -------------------------------------------- > Sorry, I need your advice. Express the integral as a repeated integral of the form > int_{x=a..b}[int_{y=L(x)..U(x) f(x,y) dy] dx. By drawing a diagram, > figure out what are a, b, L(x) and U(x). Then do the inner integral, > followed by the outer integral. That would be my advice. Did you follow your own advice? I had thought of asking Ray that very same question myself. > I tried to follow that approach. The > problem (at least for me) is the do the inner integral part. I suppose that there's an easier method, but that approach does work in Mathematica: In[5]:= Integrate[Sin[(y - 2*x)^2], {y, 0, 2*x + 1}] Out[5]= Sqrt[Pi/2]*(FresnelS[Sqrt[2/Pi]] + FresnelS[2*Sqrt[2/Pi]*x]) In[6]:= Integrate[%, {x, -1/2, 0}] Out[6]= 1/2*Sin[1/2]^2 In[7]:= TrigReduce[%] Out[7]= 1/4*(1 - Cos[1]) David === Subject: Re: Calculus with double integral.. > R : the interior and boundary of the triangle with (0, 0), (-1/2, > 0), (0, 1). > Double int{R} sin{(y - 2x)^2} dA = ? > Answer : (1/4)(1 - cos1) > -------------------------------------------- > Sorry, I need your advice. Express the integral as a repeated integral of the form > int_{x=a..b}[int_{y=L(x)..U(x) f(x,y) dy] dx. By drawing a diagram, > figure out what are a, b, L(x) and U(x). Then do the inner integral, > followed by the outer integral. That would be my advice. Did you follow your own advice? I had thought of asking Ray that very same question myself. I tried to follow that approach. The > problem (at least for me) is the do the inner integral part. I suppose that there's an easier method, but that approach does work in > Mathematica: In[5]:= Integrate[Sin[(y - 2*x)^2], {y, 0, 2*x + 1}] > Out[5]= Sqrt[Pi/2]*(FresnelS[Sqrt[2/Pi]] + FresnelS[2*Sqrt[2/Pi]*x]) In[6]:= Integrate[%, {x, -1/2, 0}] > Out[6]= 1/2*Sin[1/2]^2 In[7]:= TrigReduce[%] > Out[7]= 1/4*(1 - Cos[1]) Now the easy way: Integrate first along line segments parallel to y = 2x, specifically, along y = 2x + c. In[8]:= Integrate[Sin[c^2], {x, -c/2, 0}] Out[8]= 1/2*c*Sin[c^2] In[9]:= Integrate[%, {c, 0, 1}] Out[9]= 1/2*Sin[1/2]^2 David === Subject: Re: Calculus with double integral.. > R : the interior and boundary of the triangle with (0, 0), (-1/2, 0), > (0, 1). > Double int{R} sin{(y - 2x)^2} dA = ? > Answer : (1/4)(1 - cos1) > -------------------------------------------- > Sorry, I need your advice. > Express the integral as a repeated integral of the form > int_{x=a..b}[int_{y=L(x)..U(x) f(x,y) dy] dx. By drawing a diagram, > figure out what are a, b, L(x) and U(x). Then do the inner integral, > followed by the outer integral. That would be my advice. > Did you follow your own advice? I had thought of asking Ray that very same question myself. > I tried to follow that approach. The > problem (at least for me) is the do the inner integral part. I suppose that there's an easier method, but that approach does work in > Mathematica: In[5]:= Integrate[Sin[(y - 2*x)^2], {y, 0, 2*x + 1}] > Out[5]= Sqrt[Pi/2]*(FresnelS[Sqrt[2/Pi]] + FresnelS[2*Sqrt[2/Pi]*x]) In[6]:= Integrate[%, {x, -1/2, 0}] > Out[6]= 1/2*Sin[1/2]^2 I did the same thing. > In[7]:= TrigReduce[%] > Out[7]= 1/4*(1 - Cos[1]) This I had not done, but I used the fact that 1 - cos(2x) = cos(x)^2 + sin(x)^2 - (cos(x)^2 - sin(x)^2) = 2sin(x)^2. Jose Carlos Santos === Subject: Is aleph smaller than beth? Let S be a set. Let us not assume AxC, CH, nor GCH. But just a nanosec. GCH --> AxC, so we can skip GCH. Isn't this theorem 'obvious'? That is clearly, some xi with |S| = aleph_xi. How about this theorem? Is it a theorem? some xi with |S| <= beth_xi. beth_0 = aleph_0 beth_(xi+1) = 2^beth_xi beth_eta = sup{ beth_xi | xi < eta }, eta limit ordinal Riddle of the day. What comes after beth and is it bigger? ---- === Subject: Re: Is aleph smaller than beth? > Let S be a set. Let us not assume AxC, CH, nor GCH. Isn't this theorem 'obvious'? That is clearly, > some xi with |S| = aleph_xi. It's 'obvious' perhaps, but it's not a theorem without AxC. It asserts that there exists a bijection between S and some well-ordered cardinal aleph_xi, and so S can be well-ordered. Hence it is equivalent to AxC. - Tim === Subject: Merry Christmas 6 posting-account=AcTHNQoAAADG_Se8DlyOgUX_HZB4AEk6 AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.1 Safari/525.18,gzip(gfe),gzip(gfe) This specification is not copyrighted or patented. I intend it to be public domain for free distribution. The data encryption technique is useful for both small and large data streams, having no bloat. Because every bit of the input is processed individually, it may be slow for gigabyte archives. It requires a large number of random number generator seeds, so consult the specification Merry Christmas 2 on Google group Lonnie Courtney Clay for how to obtain them from a single password string. 1) There are only two states possible for a bit scramble - pass through or complemented. Generate an array of 64k bit flags indicating which operation is to be done on the input bit when the index of the bit flag is selected. 2) Generate 16 arrays of 16 bits by 64k entries which give either the index of the array 1) or which are 16 times one bit of the concatenated index to be taken from each of the 16 arrays, depending upon a user option. If one bit from each to be concatenated to form an index is the option, then the order in which the 16 array's bits are used when generating the index is determined below. If the bits are concatenated, then the order in which the array's bits are used may be from right to left or left to right depending upon a user option. If the index is taken from each array directly, then the order in which the indexes are pulled is determined below. 3) Generate an array of 64k entries each of which is 16 bytes. The 16 bytes are used to specify which of the 16 arrays in 2) above are to be used for the next bit, or to specify the order of the arrays in which the bits are concatenated to form an index. A user option specifies whether the 16 byte values ranging from 0-15 are to be generated randomly without duplication, or generated randomly with duplication allowed. 4) Generate an array of 64k random number seeds which are used to produce the arrays 1) 2) 3) above. When a random number seed has a short life and begins to repeat without recovery, switch to another seed. When a seed is switched, the index of the next seed to be used can be simply incremented, or selected by some more esoteric technique depending upon user options. For example rather than simply incrementing the index, the next index may be determined from the value of the discarded seed in some manner. 5) Generate an array of 64k 16 bit numbers from a master seed list which indexes the start location in 4) for pulling a random number seed. As the input bit stream is processed, periodically increment the index to this array to prevent reuse of the arrays in predictable ways. For example, the array 3) above processes one megabit without repetition. For multiple gigabyte archives the reuse of a coding scheme would be unacceptable. With 64k start indexes available, the scope without repetition is increased to 64 gigabytes, which should be sufficient. The process of switching to the next start index could at user option involve regenerating the arrays 1) 2) 3) from 4). Composing this document took one hour. I am posting it early because I have been getting ready for a July 4 post for a couple of weeks now. I see nothing further to add to make this specification more robust, so here is the July 4 post early, on Father's Day. The next two posts will be on Labor Day September 1 and Christmas. I have received no gifts or any other form of compensation for any of this Merry Christmas series. If you have found the ideas to be useful, then please send a check to me at 3395 Harrell Road, Arlington TN 38002-4261. Lonnie Courtney Clay Laughing Crazy Coot TARZAN Chic Logo Guy === Subject: Axiom of Foundation posting-account=jNR8OgoAAABgTHQuvrwdZAvYKYRyZzQh Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) We work in Zermelo-Fraenkel set theory. Consider the collection S = {x | there is an epsilon formula P(y) such that x = {y | P(y)} }. Then S is a set (being countable). But S must belong to S which contradicts the axiom of foundation which appears absurd. Where exactly is the flaw in this argument? Any help would be appreciated. === Subject: Re: Axiom of Foundation > We work in Zermelo-Fraenkel set theory. Consider the collection S = {x > | there is an epsilon formula P(y) such that x = {y | P(y)} }. Then S > is a set (being countable). But S must belong to S which contradicts > the axiom of foundation which appears absurd. Where exactly is the > flaw in this argument? Any help would be appreciated. The problem is when you try to properly articulate what there is an epsilon formula. If you push these thoughts to their limit, you will end up rediscovering a form of Goedel's Theorem. === Subject: Discrete math with fish... Hello teacher~ There are 5 fishes(A, B, C, D, E) in a fish bowl. A eats B. B eats C. C eats D. D eats E and A. Put 5 fishes on distinct 4 fish bowl(1, 2, 3, 4) with PEACE(not eat). Find the number of cases. (Empty fish bowl permit.) Answer : 252 -------------------------------------------------- Sorry. I need your advice. === Subject: Re: Discrete math with fish... > Hello teacher~ There are 5 fishes(A, B, C, D, E) in a fish bowl. A eats B. > B eats C. > C eats D. > D eats E and A. Put 5 fishes on distinct 4 fish bowl(1, 2, 3, 4) with PEACE(not eat). > Find the number of cases. > (Empty fish bowl permit.) Answer : 252 -------------------------------------------------- > Sorry. I need your advice. All this plus calculus and Lebesgue integration? Busy busy busy. === Subject: Re: Discrete math with fish... posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > Hello teacher~ There are 5 fishes(A, B, C, D, E) in a fish bowl. A eats B. > B eats C. > C eats D. > D eats E and A. How can this happen? If A eats B and B eats C, B must first eat C and then get eaten by A. Since C eats D, C must first eat D before being eaten by B. The first three statements thus imply that A is the last fish in the bowl after everybody has eaten (although poor D never eats anything). The fourth statement contradicts all this. Perhaps your verb eat does not really mean 'consume', but something else, like A knows B (these are intelligent fish). Put 5 fishes on distinct 4 fish bowl(1, 2, 3, 4) with PEACE(not eat). What does this mean? It seems to make no sense. R.G. Vickson > Find the number of cases. > (Empty fish bowl permit.) Answer : 252 -------------------------------------------------- > Sorry. I need your advice. === Subject: Re: Discrete math with fish... > Hello teacher~ > There are 5 fishes(A, B, C, D, E) in a fish bowl. > A eats B. > B eats C. > C eats D. > D eats E and A. How can this happen? If A eats B and B eats C, B must first eat C and > then get eaten by A. Since C eats D, C must first eat D before being > eaten by B. The first three statements thus imply that A is the last > fish in the bowl after everybody has eaten (although poor D never eats > anything). The fourth statement contradicts all this. Perhaps your > verb eat does not really mean 'consume', but something else, like A > knows B (these are intelligent fish). These are the property of fishes. This is not event. > Put 5 fishes on distinct 4 fish bowl(1, 2, 3, 4) with PEACE(not eat). What does this mean? It seems to make no sense. I already gave the answer. Put 5 fishes on distinct 4 fish bowl(1, 2, 3, 4) without eating event. === Subject: Re: Discrete math with fish... posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > Hello teacher~ > There are 5 fishes(A, B, C, D, E) in a fish bowl. > A eats B. > B eats C. > C eats D. > D eats E and A. How can this happen? If A eats B and B eats C, B must first eat C and > then get eaten by A. Since C eats D, C must first eat D before being > eaten by B. The first three statements thus imply that A is the last > fish in the bowl after everybody has eaten (although poor D never eats > anything). The fourth statement contradicts all this. Perhaps your > verb eat does not really mean 'consume', but something else, like A > knows B (these are intelligent fish). These are the property of fishes. > This is not event. I don't know what this means. Does your word these refer to your original question, or to my response? Which properties are you referring to? When you say that this is not event, what do you mean? (What is this, and why is it not an event?) I *really am* trying to understand your question, but the way you stated it seems to lack meaning. > Put 5 fishes on distinct 4 fish bowl(1, 2, 3, 4) with PEACE(not eat). What does this mean? It seems to make no sense. I already gave the answer. But I did not understand your answer. Put 5 fishes on distinct 4 fish bowl(1, 2, 3, 4) without eating event. So, are you sating that the scenario is really something like the one I mentioned, where you could say that A knows B, etc? In that case, what are you trying to determine? Are you asking for the number of graphs of some type? For example, in your original question you get a graph with vertices A,...,E, with an arc between two vertices i and j if i eats j or j eats i. It is an undirected graph if you don't care which fish eats the other, but a directed graph if the eat relation goes one way. In your case you had A -> B -> C -> D, D -> A and D -> E. This has one isolated node (E), while the others form a 4-cycle. Is that the defining property you want? R.G. Vickson === Subject: Re: Discrete math with fish... > Hello teacher~ > There are 5 fishes(A, B, C, D, E) in a fish bowl. > A eats B. > B eats C. > C eats D. > D eats E and A. > How can this happen? If A eats B and B eats C, B must first eat C > ... > I already gave the answer. But I did not understand your answer. > Have you really not understood, or is it a trial to make mina's posts more understandable ? What was required was clearly to find all arrangements of fishes in bowls such that : { A and B are not in the same bowl (otherwise A would eat B) and B and C are not in the same bowl (otherwise B would eat C) and C and D are not in the same bowl (otherwise C would eat D) and D and E are not in the same bowl (otherwise D would eat E) and D and A are not in the same bowl (otherwise D would eat A) } That's all. Not really a question of graph theory, even if graph theory may help. But the question was just how many such arrangements. -- Philippe Ch., mail : chephip+news@free.fr site : http://mathafou.free.fr/ (recreational mathematics) === Subject: Re: Discrete math with fish... > Hello teacher~ There are 5 fishes(A, B, C, D, E) in a fish bowl. A eats B. > B eats C. > C eats D. > D eats E and A. Put 5 fishes on distinct 4 fish bowl(1, 2, 3, 4) with PEACE(not eat). > Find the number of cases. > (Empty fish bowl permit.) Answer : 252 -------------------------------------------------- > Sorry. I need your advice. Why? Is there a problem in understanding the problem, or is it just that you don't like doing it yourself? In the first case consider 1=AC 2=BD 3=empty 4=empty How about peace? Well there is peace because A and C are not in an eating relation and neither are B and D. A not so spectacular variation of this fishs-in-a-bowl is e.g. 1=empty 2=BD 3=AC 4=empty. My advice: find more combinations and count them. Compare the result to 252. If equal, you are done. Otherwise there will be some more to be found. Rainer Rosenthal r.rosenthal@web.de === Subject: Re: Discrete math with fish... > Hello teacher~ > There are 5 fishes(A, B, C, D, E) in a fish bowl. > A eats B. > B eats C. > C eats D. > D eats E and A. > Put 5 fishes on distinct 4 fish bowl(1, 2, 3, 4) with PEACE(not eat). > Find the number of cases. > (Empty fish bowl permit.) > Answer : 252 > -------------------------------------------------- > Sorry. I need your advice. Why? Is there a problem in understanding the problem, or is it > just that you don't like doing it yourself? In the first case consider 1=AC 2=BD 3=empty 4=empty > How about peace? Well there is peace because A and C are > not in an eating relation and neither are B and D. > A not so spectacular variation of this fishs-in-a-bowl is e.g. > 1=empty 2=BD 3=AC 4=empty. My advice: find more combinations and count them. Compare the > result to 252. If equal, you are done. Otherwise there will be > some more to be found. Your advice has astonishing power even if no details. Anyway, I tried again. and I succeeded. ACE / BD / empty / empty ==> 12 AC / B / D / E ==> 4! AC / BD / E / emtpy ==> 4! AC / BE / D / emtpy ==> 4! AD / B / C / E ==> 4! AD / BE / C ==> 4! AD / CE / B ==> 4! BD / A / C / E ==> 4! BD / CE / A ==> 4! BE / A / C / D ==> 4! CE / A / B / D ==> 4! so, 12 + 4!*10 = 12 + 240 = 252. === Subject: Re: Discrete math with fish... > Your advice has astonishing power even if no details. Well, at least enough details to help you, right? > Anyway, I tried again. and I succeeded. Good, but see my remark at the end of this posting. ACE / BD / empty / empty ==> 12 ... > You are welcome. I didn't yell, but I wanted you to start, and it seemed as if you were like a lonesome fish in a bowl :-) One more advice: it would be nice if you stated something like: 1. there is no peaceful 4-set or 5-set 2. there is only one peaceful triplet: ACE 3. the only peaceful pairs are: AC, AD, AE, BD, BE, CE In doing so, you prepare yourself and your reader for what you list up. And - you are able to check your solution. I didn't go very deep into your list, but I wonder why you left out the peaceful pair AE. Somethings seems to be wrong here. Rainer Rosenthal r.rosenthal@web.de === Subject: Some questions on non-measurable sets posting-account=jNR8OgoAAABgTHQuvrwdZAvYKYRyZzQh Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) (1) Let r be a positive real number. Give an example of a non measurable set whose outer measure is r. (2) For every natural number n, determine if there exists a collection {A(r) | r = 1, 2,..., n} of pairwise disjoint non measurable subsets of [0, 1] such that m*(U A(r)) = Sum m*(A(r))? (3) Is there a collection {A(n) | n = 1, 2, ...} of pairwise disjoint non measurable sets in [0, 1] such that m*(U A(n)) = Sum m*(A(n))? Any help would be appreciated. === Subject: Re: Some questions on non-measurable sets posting-account=AdyLXQoAAABgRay99CKv1O8Y_7jjivwq Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > (1) Let r be a positive real number. Give an example of a non > measurable set whose outer measure is r. (2) For every natural number n, determine if there exists a collection > {A(r) | r = 1, 2,..., n} of pairwise disjoint non measurable subsets > of [0, 1] such that m*(U A(r)) = Sum m*(A(r))? (3) Is there a collection {A(n) | n = 1, 2, ...} of pairwise disjoint > non measurable sets in [0, 1] such that m*(U A(n)) = Sum m*(A(n))? Any help would be appreciated. I don't have a lot of time (I'm displaced because of flooding) but here are things I've posted about in the past that are more than you asked for: Nikolai N. Lusin and Waclaw Sierpinski, Sur une d.8ecomposition d'un intervalle en une infinit.8e non d.8enombrable d'ensembles non mesurables [On a decomposition of an interval into a nondenumerably many nonmeasurable sets], Comptes Rendus Acad.8emie des Sciences (Paris) 165 (1917), 422-424. [JFM 46.0294.01] [available on-line] http://www.emis.de/cgi-bin/JFM-item?46.0294.01 They prove that the unit interval [0,1] can be partitioned into c = 2^(aleph 0) sets that are pairwise disjoint and which have outer Lebesgue measure equal to 1. However, I think the result you want (countably many nonmeasurable, with no conditions placed on their outer measures) is much easier. Think about the Vitali set construction, and how certain translates of the set constructed will be disjoint from each other. For the third question, use sets A and B, where A has outer measure 1/2 and is a subset of [0, 1/2] and B has outer measure 1/2 and is a subset of (1/2, 1]. As for your first question, can you construct a set with outer measure r in the interval [0, r]? See my post on Bernstein sets: [correction] Dave L. Renfro === Subject: Re: Some questions on non-measurable sets posting-account=jNR8OgoAAABgTHQuvrwdZAvYKYRyZzQh Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > (1) Let r be a positive real number. Give an example of a non > measurable set whose outer measure is r. (2) For every natural number n, determine if there exists a collection > {A(r) | r = 1, 2,..., n} of pairwise disjoint non measurable subsets > of [0, 1] such that m*(U A(r)) = Sum m*(A(r))? (3) Is there a collection {A(n) | n = 1, 2, ...} of pairwise disjoint > non measurable sets in [0, 1] such that m*(U A(n)) = Sum m*(A(n))? Any help would be appreciated. Correction in (2): It is also required that Union r = 1 to n (A(r)) be non-measurable. === Subject: Re: Some questions on non-measurable sets posting-account=jNR8OgoAAABgTHQuvrwdZAvYKYRyZzQh Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) (1) Let r be a positive real number. Give an example of a non > measurable set whose outer measure is r. (2) For every natural number n, determine if there exists a collection > {A(r) | r = 1, 2,..., n} of pairwise disjoint non measurable subsets > of [0, 1] such that m*(U A(r)) = Sum m*(A(r))? (3) Is there a collection {A(n) | n = 1, 2, ...} of pairwise disjoint > non measurable sets in [0, 1] such that m*(U A(n)) = Sum m*(A(n))? Any help would be appreciated. Correction in (2): It is also required that Union r = 1 to n (A(r)) be > non-measurable. Please ignore problems (2) and (3). They turned out to be quite trivial. === Subject: Re: Some questions on non-measurable sets (1) Let r be a positive real number. Give an example of a non > measurable set whose outer measure is r. (2) For every natural number n, determine if there exists a collection > {A(r) | r = 1, 2,..., n} of pairwise disjoint non measurable subsets > of [0, 1] such that m*(U A(r)) = Sum m*(A(r))? (3) Is there a collection {A(n) | n = 1, 2, ...} of pairwise disjoint > non measurable sets in [0, 1] such that m*(U A(n)) = Sum m*(A(n))? Any help would be appreciated. Correction in (2): It is also required that Union r = 1 to n (A(r)) be > non-measurable. Please ignore problems (2) and (3). They turned out to be quite > trivial. For (1), note that m*(tE) = tm*(E) for all E and t >= 0. === Subject: Re: car's trajectory posting-account=jMsxagoAAAC74IpZ9bM5CVr1u8LkT2oO Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > Consider a vehicle of axle-to-axle length L, > and left to right wheel separation W (though > I don't believe this matters). Assume a X-Y co-ordinate plane, the origin > located at the center of the rear axle. What is the car's forward trajectory, > if the front wheels rotate left at angle U? > What is its reverse trajectory? What if the car has front wheel drive? Why does one park into a space by backing > up, rather than forward? ** BONUS CREDIT ** Use your answer above, to solve the parallel > parking problem. Mark How to Modify Your Car Cheaply to RUN ON WATER THE PROBLEMS: Foolish petrol prices, petrol battles, pollution and global climate change. You know what I'm referring to. THE BAD NEWS: One likely way out, that you may have heard of, is water automobiles. As distinct from water hybrids, water powered automobiles are yet to be rolled out of the assembly plants. You could get a minimum of 10 strategies of water powered automobiles to try out. Regrettably those designs are too costly to put together, are very knotty, and absolutely unpredictable. THE GOOD NEWS: Widespread development and experimentation by numerous back yard engineers has proven one method FEASIBLE. With this method you can run your traditional automobiles PARTIALLY ON WATER (that's a Water HybridØ) with little or no modifications. ¢ You should be able to increase your gas efficiency by as much as 59% (Toyota 99) or even 70% (Cadillac 99). ¢ You could drastically cut down discharges and help reduce Global Warming. ¢ You can improve efficient burning of fuel and consequently lessen wear and tear, engine clatter, high temperature in the engine, carbon deposits, etc. You see, you have just hit upon a unique piece of information. As there is a small number of persons on the World Wide Web who will not sell you products, but rather when you become a member of a group or club, you will will gain knowledge of how to make these stuffs yourself at home. A number of them will even provide you a COMPLIMENTARY HYDROGEN BOOSTER SYSTEM, fully put together set to go into your vehicle, so you can authenticate its practicability and have a copy for replication. Again, this is a COMPLIMENTARY GIFT for you, for becoming a member and a researcher You could make use of this mock system to switch nearly any vehicle into an trial, wholly functional WATER HYBRID in one slow day, with the aid of simple tools at home. Back in I knew nothing about automobiles, I had no garage and no special tools, yet I installed one un-aided within a quarter of an hour. This was in an era when all I had was half a page of instructions. Nowadays you can obtain complete informational products or ebooks, packed with guidelines and suggestions. There are also support groups to assist you with any mechanical, electrical or tuning trouble. These new groups have labored hard to make it SIMPLE FOR YOU. And it is the influence of the group that makes your path so much easier and smoother. IT GETS EVEN BETTER: You will also be skilled in how to become a home manufacturer of numerous models of hydrogen generating system. Again, basic skills and easy tools is all that is need to replicate these systems for your friends, family or clients. PAY NO ROYALTIES OR LICENSE FEE - you keep all the cash! When you get into one of these exceptional clubs or groups, you can discover answers to your questions: 1. Is this really practical? How come it's so easy? 2. Why didn't anyone tell me about it before? 3. could water technology do for me? 4. Why should I become a member of a group/club? Soìyou will start to realize why water hybrid (water car / watercar) secrets have been unknown until today. About the author To get involved in knowing all there is to know and in learning how to POWER YOUR CAR WITH WATER, visit http://sampvasant.water4gas.hop.clickbank.net/?tid=6666 === Subject: Re: car's trajectory /BAH === Subject: Re: The Essential History of Special Relativity The genesis of the theory of relativity was a long process that > involved three major players and their critical reactions to the > electrodynamics of moving bodies. > Lorentz made a key step when he sought to develop a mechanics that > would obey the principle of relativity and Maxwell's equations. > Lorentz exploited the invariance properties of the fundamental > equations for the interaction between electrons and fields, and thus > accounted for the absence of effects of the motion of the earth > through the ether, but only to a certain approximation. > Poincar.8e made this absence of effects a general postulate and > elevated the principle of relativity even higher than Lorentz did. He > put the Lorentz transformations into a perfect form, discovered their > group properties and gave them a physical interpretation. He used > these transformation equations to reveal the perfect invariance of > the electromagnetic equations and to create a Lorentz-invariant > theory of gravity. > Einstein made Poincar.8e's theory completely symmetric by placing the > space and time determinations in any two inertial systems on exactly > the same footing. > But this was done before by Poincar.8e, as Lorentz also recognized. > Note also that was Poincar.8e the first who introduced the invariant > s^2 = c^2 t^2 - x^2 > which naturally implies that space and time may be in same footing. > He also simplified relativity by eliminating the ether and by > declaring two previously accepted results were fundamental > transformation. > http://www.everythingimportant.org/relativity/directory.htm > The next significant development in the history of relativity > occurred when I eliminated everything from relativity that was not > amenable to experimental verification. This was achieved by > specifying an irreducible axiom set that produces the least confusion > for beginners, which is the set of absolute minimum requirements for > a relativistic theory to exist. My theory derives the Lorentz > transformation without using Einstein's first or second postulate. > http://www.everythingimportant.org/relativity/special.pdf > The reductions and simplifications created in this theory are both > dramatic and unpleasant. The consequences are severe in that the > whole edifice of special relativity has been reduced to a near > tautology, which requires more work. The theory states that all the > laws of physics may be divided into two distinct categories. There > are physical laws that are the same in all frames of reference and > there may be laws that aren't. > You do not give a definition for same, therefore I have a doubt about > that you really mean. Where did I use the word same or the concept of sameness? I don't know > what you mean. You said (blockquote The theory states that all the laws of physics may be divided into two distinct categories. There are physical laws that are the *same* in all frames of reference and there may be laws that aren't. ) I want to know what criteria you use to decide if two laws are the same or aren't. > Are you noticing the difference between invariant and covariant laws? I always thought that physicists used the words invariant and > covariant as synonyms, so again, I don't know what you mean. invariant (rather than covariant) actions. http://en.wikipedia.org/wiki/Invariant_%28physics%29 Some authors use the terms special covariance for the former case and general covariance case for the latter but in any case the two cases are differentiated. -- Center for CANONICAL |SCIENCE) http://canonicalscience.org === Subject: Re: The Essential History of Special Relativity posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) On Jun 15, 8:04 am, Juan R. Gonz.87lez-.8dlvarez > You said (blockquote > The theory states that all the laws of physics may be divided into two > distinct categories. There are physical laws that are the *same* in all > frames of reference and there may be laws that aren't. > ) I want to know what criteria you use to decide if two laws are the same > or aren't. that I have to rewrite my interpretation. I propose the following revision, if you believe it's sufficiently clear: The theory states that all the laws of physics may be divided into two distinct categories. There are physical laws that are frame independent and there may be physical laws that aren't. A frame is an inertial frame of reference. A law is a mathematical equation that has a physical interpretation. Laws of interest consist of equations containing physical constants and parameters (such as rest mass, spin, charge, position, time, velocity, acceleration, etc) or other abstract or measurable quantities (such as quantum amplitudes, magnetic and gravitational field strengths etc). A physical law is frame independent if the equation describing some aspect of motion or the interactions of matter and energy in spacetime is true in all inertial frames of reference. > Are you noticing the difference between invariant and covariant laws? Yes, that's what I'm referring to, if I understand the meaning of a covariant law. If so, then all the laws the physics in the original relativity theory of Lorentz and Poincar.8e are covariant laws whereas, in Einstein's interpretation, they are invariant. > I always thought that physicists used the words invariant and > covariant as synonyms, so again, I don't know what you mean. invariant (rather than covariant) actions. http://en.wikipedia.org/wiki/Invariant %28physics%29 That link doesn't define what is meant by a covariant action but I assume it means an action that is nearly invariant or possibly completely invariant if the action can be supplemented with appropriate gauge-conditions. http://xxx.lanl.gov/abs/physics/0610154 I know that general covariance in physics is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations. > Some authors use the terms special covariance for the former case and > general covariance case for the latter but in any case the two cases > are differentiated. I can see the distinction between special and general covariance in the relativity theory of Lorentz and Poincar.8e versus Einstein's theory when explained in terms of a physical distinction, but I still don't see a mathematical difference in that instance. > Center for CANONICAL |SCIENCE) http://canonicalscience.org Shubee http://www.everythingimportant.org/relativity/special.pdf === Subject: Re: The Essential History of Special Relativity Are you noticing the difference between invariant and covariant laws? > Covariance: {T^(a...)_(b...); g_ab; f(g_ab)} --> {t^(m...)_(n...); y_mn; h(y_mn)} Invariance: {T^(a...)_(b...); g_ab; f(g_ab)} --> {phi T^(a...)_(b...); g_ab; f(g_ab)} with phi an isometry. Some authors call general covariance to the former and special covariance to latter. -- Center for CANONICAL |SCIENCE) http://canonicalscience.org === Subject: Re: The Essential History of Special Relativity > On Jun 13, 10:05 am, harry Harald, please cite an internet reference for that. > Instead I have the original papers which state and acknowledge that > (Lorentz 1904, Poincare 1905). > Einstein made Poincar.8e's theory completely symmetric by placing the > space and time determinations in any two inertial systems on exactly > the same footing. > That happens to be already the case with the Lorentz transformations, > I need to edit my remark. .8bEinstein made Poincar.8e's theory completely > symmetric by putting space and time in any two inertial systems on > exactly the same footing..8a > Your comment is now false. > No, why? The Lorentz transformations comply to Poincare's PoR, according > to which the laws of physics are the same in any inertial frame. Poincar.8e's theory already had the all the necessary mathematical > symmetry that one needs for real physics. But Einstein complicated > Poincar.8e's theory by insisting on an additional unobservable symmetry > where the meaning of time and distance flip-flops back and forth, > depending on the observer. That simply doesn't happen with the > assumption of an unobservable absolute frame of reference. In this > sense, I strongly believe that Einstein complicated Poincar.8e's theory > by insisting on the unobservable. Sorry I can't parse that.... > The innovation in my approach is that I'm really going back to > Poincar.8e's easier to understand theory but in my minimalist approach > I'm not presupposing that an unobservable absolute frame of reference > exists or doesn't exist. Which is what Einstein did, as quoted. > In that sense, I believe that I've created a > theory that is more fundamental than Poincar.8e's original relativity > and Einstein's revision. At first sight that can't be right: Already Newton made the same assumptions (his first law and the PoR), but he used the Galilean equations. Thus it appears that either you or Newton made a mistake. > The next significant development in the history of relativity > occurred > when I eliminated everything from relativity that was not amenable > to > experimental verification. > Huh? Everything above IS based on experiments. > That.89s false. No experiment proves the nonexistence of an absolute > frame of reference. > SRT doesn't contain such a postulate, just the PoR which was by then a > matter of observation. Of course you can argue that not all possible > observations can be tested, but that is true for any law of physics. Poincar.8e believed in an unobservable, absolute frame of reference. > And, of course, Poincar.8e originated the PoR. Are you saying that the > PoR in special relativity, as taught today, is identical to the PoR in > Poincar.8e.89s original relativity? I disagree with the concept of modifying a theory while pretenting that what one teaches dates from the time of inception. Apart of that, the idea of different PoR concepts was unheard of, that's a later invention. Any perceived difference between the PoR of Einstein and Poincare is a splitting of hairs: the variations in formulations by each of them were greater than the differences between the two. > Even if Einstein.89s PoR is identical to the PoR in Poincar.8e.89s original > relativity and even though physicists today might not have declared > the nonexistence of an unobservable, absolute frame of reference as an > axiom clearly enough for you, they sure do presuppose its nonexistence > when explaining Einstein.89s theory. And I.89m not talking about > observations. I.89m talking about presuppositions. Later relativistis both claimed the existence or nonexistence of an ether. However, the purpose of Einstein's approach to the New Mechanics was to turn it into a principle theory which does NOT make such a preproposition, as I alread quoted - apart of the second postulate which Pauli called the true essence of the aether point of view. http://philsci-archive.pitt.edu/archive/00001385/01/9908048.pdf > I've read several papers that state that Lorentz's theory evolved from > v/c order to exactness. See, for example, > http://en.wikipedia.org/wiki/Lorentz_ether_theory > where it says: > A fundamental concept of Lorentz's theory in 1895[4] was the theorem > of corresponding states for terms of order v/c. This theorem states > that a moving observer (relative to the ether) in his .b3fictitious.8b > field makes the same observations as a resting observers in his .b3real.8b > field. This theorem was extended for terms of all orders by Lorentz > (1904)[5] and completed by Poincar.8e (1905, 1906)[6][7] and by Lorentz > (1906, 1916)[8] in order to obey the principle of relativity. > As Lorentz put it himself: > It would be more satisfactory if it were possible to show by means of > certain fundamental assumptions and without neglecting terms of one order > of magnitude or another, that many electromagnetic actions are entirely > independent of the motion of the system. Some years ago, I already sought > to frame a theory of this kind. I believe it is now possible to treat > velocity will be that it be less than that of light. I shall start from > the fundamental equations of the theory of electrons. This quote from Lorentz confirms what I.89ve said. Lorentz started > relativity by approximating the consequences to the PoR in terms of > first order effects. The phrase, .8bwithout neglecting terms of one > order of magnitude or another.8a doesn.89t admit an oversight or a > mathematical error. It simply refers to the business of making an > approximation. Of making NO approximation. > Those fundamental assumptions were really new. Probably the word many > was put there because it appeared to him that it still didn't work > perfectly. Poincare corrected the glitch and showed that Lorentz's new > theory resulted in the Lorentz transformations which achieve perfect > relativity of inertial frames. How does that contradict my opening post? Your claim only to a certain approximation. That should be: but not entirely without error. Harald === Subject: Re: The Essential History of Special Relativity > On Jun 13, 10:05 am, harry >harald.vanlintelButNotT...@epfl.ch > The next significant development in the > history of relativity occurred > when I eliminated everything from relativity > that was not amenable to > experimental verification. > Huh? Everything above IS based on experiments. ThatÍs false. No experiment proves the > nonexistence of an absolute > frame of reference. SRT doesn't contain such a postulate, I found a reference for you in wikipedia. > http://en.wikipedia.org/wiki/Special_relativity section titled > ñPostulatesî, it clearly states: It should be noted that the derivation of special > relativity depends > not only on these two explicit postulates, but also > on several tacit > assumptions (which are made in almost all theories of > physics), > including the isotropy and homogeneity of space and > the independence > of measuring rods and clocks from their past > history. (Einstein, > Fundamental Ideas and Methods of the Theory of > Relativity, 1920). The independence of measuring rods and clocks from > their past history > is not an assumption that I'm willing to make. > Then you don't grasp the mathematical model. Spatial isotropy and homogeneity imply time as a simple parameter of reversible direction. Lorentz Transformation would not be true otherwise. Thus, past history of the measuring instrument can play no role in the measurement result, or the observers could not agree on the result. Tom > Shubee > http://www.everythingimportant.org/relativity/special. > pdf === Subject: Re: The Essential History of Special Relativity posting-account=rIfu6QoAAAD5nXG3h9QEE0J3dZn1U45R Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) SR remains merely a conjecture --- an absurd one actually. æ Why is the conjecture of Lorentz invariance absurd? He doesn't understand or accept any resolution of the Twins paradox, nor does he understand when he can and cannot apply the Lorentz transformations. === Subject: n*2^k+17 posting-account=WR65RgoAAACboMXdKKYDn4ZKOJhMgvsz Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) Prove that for every natural number k there exists natural number n such that n*2^k+17 is a square of some natural number. P.S. I tried using induction however it did not work for me. === Subject: Re: n*2^k+17 posting-account=VR0DOgoAAADggPTteFeA2AkmHNhjcrDV Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > Prove that for every natural number k there exists natural number n > such that n*2^k+17 is a square of some natural number. P.S. > I tried using induction however it did not work for me. Induction will work. Suppose that the result holds at k. We need to show that the result holds at k + 1. Note that we can assume that k is greater than or equal to 3. Because the result holds at k, there is an integer x (greater than 4) such that x^2 - 17 is divisible by 2^k. Note that x is odd. If x^2 - 17 is divisible by 2^{k+1}, then we are finished. If x^2 - 17 is not divisible by 2^{k+1}, let y = x + 2^{k-1}. Then y^2 = x^2 + (x)(2^k) + 2^{2k-2}. Because k is greater than or equal to 3, the term 2^{2k-2} is divisible by 2^{k+1}. As to the rest, the remainder when x^2 is divided by 2^{k+1} is 2^k. The remainder when (x)(2^k) is divided by 2^{k+1} is also 2^k. It follows that the remainder when y^2 is divided by 2^{k+1} is 0. === Subject: Re: n*2^k+17 posting-account=WR65RgoAAACboMXdKKYDn4ZKOJhMgvsz Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) Prove that for every natural number k there exists natural number n > such that n*2^k+17 is a square of some natural number. P.S. > I tried using induction however it did not work for me. > Induction will work. Suppose that the result holds at k. We need to > show that the result holds at k + 1. Note that we can assume that k > is greater than or equal to 3. Because the result holds at k, there is an integer x (greater than 4) > such that x^2 - 17 is divisible by 2^k. Note that x is odd. If x^2 - 17 is divisible by 2^{k+1}, then we are finished. If x^2 - 17 is not divisible by 2^{k+1}, let y = x + 2^{k-1}. Then y^2 = x^2 + (x)(2^k) + 2^{2k-2}. Because k is greater than or > equal to 3, the term 2^{2k-2} is divisible by 2^{k+1}. As to the rest, the remainder when x^2 is divided by 2^{k+1} is 2^k. 2^5 | 9^2 - 17 9^2 = 1*2^6 + 17 17 != 32 It is not true that the remainder when x^2 is divided by 2^{k+1} is 2^k. > The remainder when (x)(2^k) is divided by 2^{k+1} is also 2^k. It > follows that the remainder when y^2 is divided by 2^{k+1} is 0. I also do not fallow this. If n*2^k=(x^2)-17 and y=x+2^k then y^2=(x^2)+x2^{k+1}+2^{2k} then (y^2)-17=(2^k)(n+2x+2^k) so (y^2)-17 is divisible by 2^{k+1} only if n is even and n does not have to be even. === Subject: Re: Angle of Wheel - Stumped <48533924$0$20327$9a6e19ea@news.newshosting.com> posting-account=zga2wgoAAAD_6fmi3XyA1bMyNINP0zBK WorldLynx),gzip(gfe),gzip(gfe) > There are two wheels, in a gear-like assemblage. One wheel is 5 cm > radius, the other is 10 cm radius. The large wheel is in a fixed > position, and cannot rotate. The small wheel sits on the large wheel, > at the very top of the wheel, and can traverse freely around the > circumference of the large wheel, rotating as it does so. The centre > of the small wheel is point A, and there is a line drawn straight > down from point A to point P on the circumference of the small > wheel. The centre of the large wheel is point B. > The small wheel starts its journey around the large wheel, and stops > when point A has travelled 8 cm from its origin (ie. an arc length). > What is the angle that the line AP now makes with the upward > vertical? > Solution: > Let A1 be the original position of A > Let A2 be the resting place of point A > Let t be the angle A1 B A2 > Let p the angle that the line AP now makes with the upward vertical > Arc length r*t = 8 cm (t = theta radians) > r = sum of both radii = 15 cm > Thus, t = 8/15 radians > Since the small circle rotates at twice the rate t increases, it will > have thus rotated 16/15 radians. > So the angle p is 16/15 - pi = .058 radians. > But my book gives the answer 1.542 radians. I have no idea why. Any > help greatly appreciated. > Anon The axle of wheel A is fastened to the axle of wheel B by what we can > call an arm. Now imagine that all 3 objects (A, B, and the arm) are > locked together and given a rotation, say clockwise, of your 8/15 > radian. At this point, each of the 3 parts has rotated 8/15 of a > radian clockwise. Now unlock the assembly, hold the arm fixed, and > give wheel B a counter-clockwise rotation of 8/15 radian, which > returns it to its original position. During this 8/15 radian rotation, > wheel A must turn an additional amount of twice 8/15, i.e., 16/15 > radian, again clockwise. The total rotation of wheel A is then the sum > of the original 8/15 plus the final 16/15, or 24/15 radian. This > amount (24/15, or 1.6 radian) doesn't agree with your book, so > possibly I have misunderstood your description of the setup. Or, > possibly, the book is wrong ......? HTH, Grover Hughes The answer is indeed 1.542 radians. æ Say that the center of the smaller circle travels from > its initial position A to A', an arc distance s1 (8cm). > The angle of the displacement of A to A' as observed > from the center of the larger circle, B, is then t = s1/(r1 + r2) where r1 is the radius of the larger circle, 10cm > æ æ æ r2 is the radius of the smaller circle, 5cm The smaller circle is assumed to remain in contact > with the larger circle as it moves (it rolls). æThe > arc length traversed on the larger circle by the > contact point is thus s2 = r1*t The smaller circle, in contact with the larger > circle, will have rolled this same distance along > its circumference, carrying the initial point of > contact p to position p' on the smaller circle, > an arclength of s2 on the smaller circle displaced > from the new point of contact between the two > circles. An arclength of s2 on the smaller circle > corresponds to an angle a = s2/r2 for the ankle BA'p'. æThat's the angle at the new center > of the smaller circle, A', from the center of the > larger circle to the new point p'. If the line A'p' is extended to the vertical of the > line BA (the vertical line passing through the initial > centers of the circles), and we call the intersection C, > triangle BA'C is formed. æCall the angle BCA' b. æWe > have then t + a + b = pi To solve the problem posed we want to know angle b. > Thus b = pi - a - t æ = pi - s2/r2 - s1/(r1 + r2) Simplifying: b = (pi*r2 - s1)/r2 = ~1.542- Hide quoted text - - Show quoted text - The discrepancy in the value I gave and the one Mr. Neill gave is that my value is the supplement of his, since I interpreted the problem as defining the desired angle to be the absolute angular change in the spatial position of the line AP. This value is, as I showed, 1.6 radians. Grover Hughes === Subject: another go at RAND's one million random digits table This is attempt to continue the project started by Matt Mahoney of trying to recover (or compress the published data, as in comp.compression) the original cards from the rerandomized cards or data that appear in the celebrated RAND publication: ``A Million Random Digits with 100,000 Normal Deviates. For background, cf.: Note: I think that, in the binary file that is discussed in comp.compression, there must have been something like base 10 to base 2 conversion of a 1-million decimal digit number to completely packed base 2 binary number (same number, different base... ) http://www.rand.org/pubs/monograph_reports/MR1418/MR1418.intro.pdf or Introduction, page x mentions original blocks 1 and 2: << Block 1 was produced immediately after a careful tune-up of the machine; Block 2 was produced after one month of continuous operation without adjustment.> Block 2 Chi2 statistic for odd-even is 7.0 ; for 125,000 digits, this means the frequency of odds was about 0.5037, or that the frequency of evens was about 0.5037, within Block 2. For Block 1, the Chi2 statistic for odd-even is 3.0; for 125,000 digits, we have one of (evens, odds) at a frequency in the Block of 0.5024. Those were the original blocks. Then, rerandomization was done. As Matt Mahoney discovered, the rerandomized digits, as published, exhibit a mod 2 linear dependence, with 50 bits of entropy lost. However, his alternating sum test over 20,000 vectors of 50 digits in the published Was the deck of 20,000 punched cards (after rerandomization) cut once or a few times? Was it thoroughly reshuffled? Assuming the rerandomized cards obtained from Block 2 were kept in the same relative order (no permutations, cuts in the block, etc.), then the chi2 = 7.0 statistic in odd-even should persist in the final table. That's why I computed moving averages of blocks of 125,000 consecutive digits in the presumptive (but not quite right) original digits; actually, only moving averages of odd-even distribution were done. The graph of the 125000-digit moving average is shown at: http://www.geocities.com/ezcos/randmoving125k.jpg This is only for the odd-even distribution. An average of 0.5037 is attained near digit 300,000; that would be a block from 175k to 300k digit position, approximately. On the other hand, 1 - 0.5037 = 0.4963 is never attained on the low side. The source code to do the moving average computation is copied below. David Bernier P.S. There seems to be a discrepancy in the count of digits in block 8 out of 20 for digit distribution: This appears in the Intro. as Table I on page [xi]. RAND pdf file gives 5003 '0's and 5163 '2's for block 8; the discrepancy from data from their *.zip file should be clear ... 4923 5013 4916 4951 5109 4993 5055 5080 4986 4974 4870 4956 5080 5097 5066 5034 4902 4974 5012 5009 5065 5014 5034 5057 4902 5061 4942 4946 4960 5019 5009 5053 4966 4891 5031 4895 5037 5062 5170 4886 5033 4982 5180 5074 4892 4992 5011 5005 4959 4872 4976 4993 4932 5039 4965 5034 4943 4932 5116 5070 5011 5152 4990 5047 4974 5107 4869 4925 5023 4902 5004 5092 5162 4936 5020 5069 4914 4943 4914 4946 // Block 8 , per *.zip ^5003 ^5163 in RAND pdf file 4860 4899 5138 4959 5089 5047 5030 5039 5002 4937 4998 4957 4964 5124 4909 4995 5053 4946 4995 5059 4948 5048 5041 5077 5051 5004 5024 4886 4917 5004 4958 4993 5064 4987 5041 4984 4991 4987 5113 4882 4968 4961 5029 5038 5022 5023 5010 4988 4936 5025 5110 4923 5025 4975 5095 5051 5035 4962 4942 4882 5094 4962 4945 4891 5014 5002 5038 5023 5179 4852 4957 5035 5051 5021 5036 4927 5022 4988 4910 5053 5088 4989 5042 4948 4999 5028 5037 4893 5004 4972 4970 5034 4996 5008 5049 5016 4954 4989 4970 5014 4998 4981 4984 5107 4874 4980 5057 5020 4978 5021 4963 5013 5101 5084 4956 4972 5018 4971 5021 4901 ______________________________________________________ #include long mat[20000][11]; int mat2[20000][50]; int mat3[19999][50]; int main(void) { long i, j, start, k; long a,b,c,d,e; long num; long count; int num2; float ratio; FILE *in; FILE *out1; FILE *out2; in = fopen(G:RAND55dataRand01a.txt, r); out1 = fopen(G:r125kx.dat, w); out2 = fopen(G:r125ky.dat, w); for(i=0;i<20000;i++) { for(j=0;j<11;j++) { fscanf(in,%ld, &mat[i][j]); } } fclose(in); for(i=0;i<20000;i++) { for(j=1;j<11;j++) { num = mat[i][j]; a = num/10000; num = num-a*10000; b=num/1000; num=num-b*1000; c=num/100; num=num-100*c; d=num/10; num=num-d*10; e=num; mat2[i][5*(j-1)]= (int)a; mat2[i][5*(j-1)+1]= (int)b; mat2[i][5*(j-1)+2]= (int)c; mat2[i][5*(j-1)+3]= (int)d; mat2[i][5*(j-1)+4]= (int)e; } } /*** alternating sum to recover original cards, per rules ***/ /*** mat3[][] is not computed efficiently .... ***/ for(i=0;i<19999;i++) { for(j=0;j<50;j++) { start = i+1; num2 = mat2[start][j]; for(k=0;k2499) { if((mat3[i-2500][j]%2)==1) num--; } count++; if((i>2499) &&(count%1000)==0) { ratio = ((float)num)/((float)125000); fprintf(out2, %.8fn, ratio); fprintf(out1, %ldn, count/1000); } } } fclose(out1); fclose(out2); return 0; } === Subject: Re: another go at RAND's one million random digits table [...] > P.S. There seems to be a discrepancy in the count > of digits in block 8 out of 20 for digit distribution: This appears in the Intro. as Table I on > page [xi]. RAND pdf file gives 5003 '0's and 5163 '2's for > block 8; the discrepancy from data from their > *.zip file should be clear ... 4923 5013 4916 4951 5109 4993 5055 5080 4986 4974 > 4870 4956 5080 5097 5066 5034 4902 4974 5012 5009 > 5065 5014 5034 5057 4902 5061 4942 4946 4960 5019 > 5009 5053 4966 4891 5031 4895 5037 5062 5170 4886 > 5033 4982 5180 5074 4892 4992 5011 5005 4959 4872 > 4976 4993 4932 5039 4965 5034 4943 4932 5116 5070 > 5011 5152 4990 5047 4974 5107 4869 4925 5023 4902 > 5004 5092 5162 4936 5020 5069 4914 4943 4914 4946 // Block 8 , per *.zip > ^5003 ^5163 in RAND pdf file 4860 4899 5138 4959 5089 5047 5030 5039 5002 4937 > 4998 4957 4964 5124 4909 4995 5053 4946 4995 5059 > 4948 5048 5041 5077 5051 5004 5024 4886 4917 5004 > 4958 4993 5064 4987 5041 4984 4991 4987 5113 4882 > 4968 4961 5029 5038 5022 5023 5010 4988 4936 5025 > 5110 4923 5025 4975 5095 5051 5035 4962 4942 4882 > 5094 4962 4945 4891 5014 5002 5038 5023 5179 4852 > 4957 5035 5051 5021 5036 4927 5022 4988 4910 5053 > 5088 4989 5042 4948 4999 5028 5037 4893 5004 4972 > 4970 5034 4996 5008 5049 5016 4954 4989 4970 5014 > 4998 4981 4984 5107 4874 4980 5057 5020 4978 5021 > 4963 5013 5101 5084 4956 4972 5018 4971 5021 4901 > In ``Recalculation of Table 1 from http://www.rand.org/pubs/monograph_reports/MR1418/dataqual.html , the authors write: << We corrected the second block of the data file to agree with the book and again recalculated Table 1. > << Table 1 shows one fewer zero and one more two in the eighth block than does our recalculation. > and their explanation is << We believe this discrepancy is the result of an error made in reading the digits data cards during the calculation of Table 1.> Above that, they write: << Our intent is to have the numbers presented here match those in the printed volume. > I guess the data cards aren't available; could Table I reflect the data cards correctly, and the book be inconsistent with the data cards? In any case, I'll continue to work with the *.zip file, which is expected to match the book; also, the data cards are probably lost... David Bernier === Subject: cylindrical kite posting-account=pPkmDAoAAADtD2DSupvpcaDY5UrRJwDQ 5.1),gzip(gfe),gzip(gfe) Consider a cylindrical kite with the following tubular frame: two circles joined by four straight rods of equal height. The problem is to construct the frame out of 4 units of tube so as to maximze the kite's volume. I know how to use calculus to solve the problem, but I can't even get to that point. I put: 2(2*pi*r) + 4h = 4 [two circumferences plus the four heights is 4 units in length] Then some basic algebra... 4*pi*r +4h = 4 pi*r + h = 1 h = 1 - pi*r But I can already tell here that there's a problem. r must be positive, so this formula for h would make h negative! I must be missing something pretty basic here... === Subject: Re: cylindrical kite >Consider a cylindrical kite with the following tubular frame: two >circles joined by four straight rods of equal height. The problem is >to construct the frame out of 4 units of tube so as to maximze the >kite's volume. I know how to use calculus to solve the problem, but I >can't even get to that point. I put: 2(2*pi*r) + 4h = 4 [two circumferences plus the four heights is 4 >units in length] Then some basic algebra... 4*pi*r +4h = 4 pi*r + h = 1 h = 1 - pi*r But I can already tell here that there's a problem. r must be >positive, so this formula for h would make h negative! I must be >missing something pretty basic here... No, this is fine. r is going to be a small positive number, when measured in these units. Just go ahead and maximise r^2(1 - pi*r), obtaining r = 2/(3*pi) = .2122 (approx.) and h = 1/3, so 4/3 units are used for the rods, and 8/3 units are used for the circles. -- Angus Rodgers Contains mild peril === Subject: Re: cylindrical kite posting-account=pPkmDAoAAADtD2DSupvpcaDY5UrRJwDQ 5.1),gzip(gfe),gzip(gfe) >Consider a cylindrical kite with the following tubular frame: two >circles joined by four straight rods of equal height. æThe problem is >to construct the frame out of 4 units of tube so as to maximze the >kite's volume. æI know how to use calculus to solve the problem, but I >can't even get to that point. æI put: 2(2*pi*r) + 4h = 4 æ[two circumferences plus the four heights is 4 >units in length] Then some basic algebra... 4*pi*r +4h = 4 pi*r + h = 1 h = 1 - pi*r But I can already tell here that there's a problem. ær must be >positive, so this formula for h would make h negative! æI must be >missing something pretty basic here... No, this is fine. ær is going to be a small positive number, when > measured in these units. æJust go ahead and maximise r^2(1 - pi*r), > obtaining r = 2/(3*pi) = .2122 (approx.) and h = 1/3, so 4/3 units > are used for the rods, and 8/3 units are used for the circles. -- > Angus Rodgers > Contains mild peril- Hide quoted text - both. Angus, you would make a great teacher (if you're not one already). You have a gift for responding in such a way as to not make the student *feel* stupid or demeaned, even when their question is stupid! === Subject: Re: cylindrical kite posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > Consider a cylindrical kite with the following tubular frame: two > circles joined by four straight rods of equal height. The problem is > to construct the frame out of 4 units of tube so as to maximze the > kite's volume. I know how to use calculus to solve the problem, but I > can't even get to that point. It sounds to me like you DON'T know how to use calculus to solve the problem, because if you did, you would have solved it. > I put: 2(2*pi*r) + 4h = 4 [two circumferences plus the four heights is 4 > units in length] Then some basic algebra... 4*pi*r +4h = 4 pi*r + h = 1 h = 1 - pi*r But I can already tell here that there's a problem. r must be > positive, so this formula for h would make h negative! I must be > missing something pretty basic here... Yes, you are. If r > 0 is small, 1 - pi*r is positive. It will be positive as long as r < 1/pi. R.G. Vickson === Subject: Re: Maple Wiki Vladimir, You are very welcomed. It was not originally my idea - just few people talked at Mapleprimes about a Wiki and an independent Maple patch library and I just hosted it on my web space. I hope that I won't have to work too much on it because I don't really have time and I don't even have Maple installed. Wiki is cool and I am going to add jsmath there so that formulas can be typed in Latex and displayed normally. Alec === Subject: Re: Making asymmetric cryptosystems work with assigned public keys? Please do not post inane rants about top posting. That dead horse has been sufficiently beaten. If you don;t like it then complain to microsoft. > For authentication, there is no way around it. If A encrypts > something for B and C has A's private key, then C can impersonate > A. As far as sucure communicatiosn go once authentication is > through, DH is probably the easiest PKE to impliment. > You could always generate your own public/private key pairs. Please do not top-post. Your answer belongs after (or intermixed > with) the quoted material to which you reply, after snipping all > irrelevant material. This has lost all connection to previous > messages. See the following links: -- > (newusers) > === Subject: Re: Making asymmetric cryptosystems work with assigned public keys? CWhizard said: > Please do not post inane rants about top posting. You have not demonstrated that the post was either inane or a rant. > That dead horse has been sufficiently beaten. Clearly not, since you haven't got the message yet. > If you don;t like it then complain to microsoft. When some bozo parks his Mondeo right outside your house, blocking you in, do you complain to Ford? It's hardly the manufacturer's fault if the user is too stupid to use the product properly. -- Richard Heathfield Email: -http://www. +rjh@ Google users: Usenet is a strange place - dmr 29 July 1999 === Subject: Re: Making asymmetric cryptosystems work with assigned public keys? * CWhizard: > Please do not post inane rants about top posting. That dead horse has been > sufficiently beaten. If you don;t like it then complain to microsoft. Try installing OE QuoteFix if you absolutely must use Outlook Express. Note that Outlook Express is unable to present digitally signed correctly, and mangles code lines when posting. It's not really a good idea to use OE. - Alf -- A: Because it messes up the order in which people normally read text. Q: Why is it such a bad thing? A: Top-posting. Q: What is the most annoying thing on usenet and in e-mail? === Subject: Re: Making asymmetric cryptosystems work with assigned public keys? But I'm a total wiener, and I insist on top-posting - and cross posting - anyway! > Try installing OE QuoteFix if you absolutely must use Outlook Express. -- Phlip http://assert2.rubyforge.org/assert_yin_yang.html === Subject: Re: Making asymmetric cryptosystems work with assigned public keys? fool who blames software for his own shortcomings: For authentication, there is no way around it. If A encrypts > something for B and C has A's private key, then C can impersonate > A. As far as sucure communicatiosn go once authentication is > through, DH is probably the easiest PKE to impliment. You could always generate your own public/private key pairs. > Please do not top-post. Your answer belongs after (or intermixed > with) the quoted material to which you reply, after snipping all > irrelevant material. This has lost all connection to previous > messages. See the following links: > Please do not post inane rants about top posting. That dead horse has been > sufficiently beaten. If you don;t like it then complain to microsoft. A) It wasn't inane, it's addressing a pertinent issue. B) While there are fools like you around, the horse has not been sufficiently beaten. C) It's the humans that use the programs that are doing the top-posting. it's trivial to move a cursor. Therefore the humans who top-post are the ones who should be complained to. Phil -- -- Microsoft voice recognition live demonstration === Subject: JSH: Diminishing concerns on factoring posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) For years I've had a major fear that if I had a major breakthrough with integer factorization it might have a very bad, negative impact on the global economy but now as I finish out the foundation layer of that research it is clear that my research is not impacting the world. Yeah, the world is having problems but those are its own. Ok, so what do I mean about the foundation layer? Well the simplest way to consider what my factoring research does is that while mathematicians have traditionally focused on one type of congruence I use two: 1. x^2 = y^2 mod S 2. z^2 = y^2 mod T where T is your target composite and S is what I call the surrogate, and the concept I call surrogate factoring. A little while back rather than consider S as a whole I started focusing on its prime factors, which is why I have a lot of equations with x^2 = y^2 mod p as p is an odd prime factor of S. That was a HUGE breakthrough to move from concentrating on S itself, where I had control variables I called n, alpha and k, and would puzzle over things like what was the way to pick k, to figuring out how to get p. And with the fundamental equations now known that completes the foundation layer of surrogate factoring, and removes the need for further brainstorming at this point, so there is no need for me to discuss further. Succinctly, the basic research phase on the primary layer is complete. World seems ok, as my worst fears haven't realized, so I'm kicking back, relaxing and moving to other problems as a MAJOR issue right now is monetization of the web, or so-called Web 2.0 as it's coming up as a bigger and bigger problem to get revenue flowing for creative people around the world, and my current focus is on the biggest player, so YouTube is the focus of problem solving efforts and that will take up most of my time indefinitely. Math research is done for this iteration. James Harris === Subject: Re: JSH: Diminishing concerns on factoring posting-account=OKTeIQkAAAAZk6JK1hK7-grwpoUDNy98 4334.34; Windows NT 5.1; SV1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) spider-mtc-tg10.proxy.aol.com[400C70CA] (Prism/1.2.1), HTTP/1.1 cache-mtc-ad05.proxy.aol.com[400C74C7] (Traffic-Server/6.1.5 [uScM]) > For years I've had a major fear that if I had a major breakthrough > with integer factorization it might have a very bad, negative impact > on the global economy Odd, no one else had this fear. > but now as I finish out the foundation layer of > that research it is clear that my research is not impacting the > world. ? Duh. > Yeah, the world is having problems but those are its own. Ok, so what do I mean about the foundation layer? Please, don't keep us in the dark! Well the simplest way to consider what my factoring research does is > that while mathematicians have traditionally focused on one type of > congruence I use two: 1. ?x^2 = y^2 mod S > 2. ?z^2 = y^2 mod T where T is your target composite and S is what I call the surrogate, > and the concept I call surrogate factoring. Others call it . A little while back rather than consider S as a whole I started > focusing on its prime factors, which is why I have a lot of equations > with x^2 = y^2 mod p as p is an odd prime factor of S. That was a HUGE breakthrough to move from concentrating on S itself, > where I had control variables I called n, alpha and k, and would > puzzle over things like what was the way to pick k, to figuring out > how to get p. And with the fundamental equations now known that completes the > foundation layer of surrogate factoring, and removes the need for > further brainstorming at this point, so there is no need for me to > discuss further. Does it also remove the need to factor an RSA number? Succinctly, the basic research phase on the primary layer is complete. This isn't the first time you've said this. World seems ok, as my worst fears haven't realized, Duh. > so I'm kicking back, relaxing and Getting drunk and singing to the walls? > moving to other problems as a MAJOR issue right now > is monetization of the web, or so-called Web 2.0 as it's coming up as > a bigger and bigger problem to get revenue flowing for creative people > around the world, and my current focus is on the biggest player, so > YouTube is the focus of problem solving efforts and that will take up > most of my time indefinitely. Do us a favor and don't tell us about it until you're done. Math research is done for this iteration. And yet, we'll never stop hearing about it, eh? James Harris === Subject: uniform convexity posting-account=7yBx-goAAACEMe-qKe_9wsckPLjrbKaB CLR 1.1.4322),gzip(gfe),gzip(gfe) example of a space which is strictly but not uniformly convex? I have found one in Carother's book: http://stargrave.no-ip.org:8080/pub/books/books.pdox.net/A_Short_Course_on_B anach_Space_Theory.ps page 135 excersise 10., but i don't know how to prove it. help! === Subject: OT: It's ABout Time :O Saudis To Increase Summer Oil Production http://www.cbsnews.com/stories/2008/06/14/world/main4181473.shtml === Subject: Number of triangles on a 2- D lattice posting-account=ZYmqvwoAAACYo-tXewMUCYh52xGos2rj Gecko/20080311 Firefox/2.0.0.13,gzip(gfe),gzip(gfe) hi, We have a bounded 2-D lattice , now consider the 1st quadrant (ie x>=0,y>=0, x and y are positive integers such that x <= a , y<=b). An area of a triangle is given namely A. Now we have to find how many such triangles with all vertices (both x and y cordinates positive integers) lie within the lattice (it could lie on the boundary also) Ideally what i would do is generate the NC3 coordinates where N (number of points in lattice = (a+1)*(b+1)) and check if they form a triangle of area A. We can try NC2 cordinates and generate the other x and then find y. So that is O(NC2 *min(( a+1),(b+1)) a,b < 500 lets say it is too huge as the number of points are 25*10^4, i can't even do a NC2 I tried another approach 2*Area = det { { x1,y1,1}, { x2,y2,1}, {x3,y3,1} } Now x1 = x3 + p , y1=x2 + q , x2 = x3 + r , y2 = y3 +s (p,q,r,s are integers , neg or pos). Now substituting and expanding the det gives me ps - rq = 2 * Area. One way of checking this is number of points that satisy ps =2*Area, ps = 2*Area + 1 , ps =2*area + 2 , ps = (a+1)*(b+1) I'm stuck here. Now i want to know how to calculate the number of triangles ? Can you tell me if it is the wrong approach . Are there theorems or properties that should be used instead of this approach ? === Subject: Re: Matrix elements? >Reluctant though I am to get carried away with abstract nonsense, I >feel I must add that in order to give a coherent account even of such >very practical concepts as that of a partitioned matrix, it is useful >to allow matrix elements to belong to the hom-sets of an additive (or, >I suspect, even a preadditive) category: Thus additive categories can be seen as the most general context > in which the algebra of matrices makes sense. Reluctant though I also am to delve into Mac Lane, Categories for the Working Mathematician (I need something more like Categories for the Dysfunctional Non-Mathematician!), I see this on page 194 of the first edition (in the exercises for section VIII.2): 6. (The free additive category) (a) Given an Ab-category A, construct an additive category Add(A) and an additive functor A --> Add(A) which is universal from A to an additive category. (Hint: Objects of Add(A) are n-tuples of objects of A, for n = 0, 1, ..., while arrows are matrices of arrows of A.) (b) If A is the commutative ring K, regarded as an additive category with one object, show that Add(A) is the category Matr_K described in section I.2. (Hint: Show that Matr_K has the desired universal property.) -- Angus Rodgers Contains mild peril === Subject: Re: Matrix elements? As no-one has either blown this to bits or said that it makes sense, I can't help worrying about it some more. Let me fill in some more details, and add a remark and a question. The remark is that, just as hagman mentioned that matrix entries might belong to a semiring, so it is enough for the hom-sets in the category to be commutative semigroups, written additively - we still get something reasonably looking like matrix algebra. Specialising rather then generalising, the question is whether, when P is an abelian category, so is TP. The definition of abelian categories doesn't look too complicated, so I might be able to plod through this - but does anyone already know the answer? If the answer is yes, then it is presumably a well-known construction (albeit, for all I know, a useless one). N.B. Buried in the mess below is an apology, which is worth giving in advance: *(In fact, I've definitely buggered up the ordering, because - I don't as a rule think about category theory at all! - all along I've been taking composition as a function hom(A, B) x hom(B, C) --> hom(A, C), (f, g) |--> f o g --- which is the wrong bloody way around! Sorry! I'll press ahead in the same misguided direction for the moment, but it all needs to be rewritten - unless somebody gives me a reference to where it's all been written out properly already.) If this isn't all already written out properly somewhere, AND if it's not a total waste of time (someone will surely tell me if it is), I'll take it as a project to (S-L-O-W-L-Y, as I have other things to learn) LaTeX up all the details, rather than spewing them out semi-readably in ASCII into sci.math like this. (That's sort of another apology!) >I have a horrible feeling that I would end up writing something about >a natural transformation between two functors of the category of all >[small] preadditive categories to itself [...] You see, if C is that category, then T: C -> C is a functor, where, >for any preadditive category, P, an object of TP is a finite family >(infinite matrices can also be considered in a less general context >...) (a_i)_{i in I} of objects in P, and an arrow in TP: f: (a_i)_{i in I} --> (b_j)_{j in J} is a finite family of arrows in P, (f_{i,j})_{(i,j) in IxJ}, where: f_{i,j}: a_i --> b_j in P (or it might be better to do it the other way around, I don't know). Of course, the identity arrow on the object a = (a_i)_{i in I} is the family of identity arrows 1(a) = (1(a_i): a_i --> a_i)_{i in I}. >Composition of arrows in TP is defined by matrix multiplication: (f o g)_{i,k} = sum_{j in J} f_{i,j} o g_{j,k} [in hom(a_i, c_k)] where, of course, g: (b_j)_{j in J} --> (c_k)_{k in K} in TP. (Fog, indeed!) Without actually writing out the details, I am fairly sure there are >natural transformations I --> T and ToT --> T ... (etc., etc., etc.) I'll leave these messy details for later (or for someone else!). I don't even have a list of what has to be verified - I'll either have to dig out that book by Manes on algebraic theories, or else look up monads (or perhaps triples, whatever they are - oh, OK, a triple IS a monad - duh!). Perhaps someone will tell me what I should be doing (so long as it's not rude!). :-) >When P is an additive category, this presumably relates in some nice >(but to me at the moment, horrible!) way to the biproduct considered >(assuming it can be!) as a natural transformation T --> I. I don't know what use this might be, but I should at least check that it makes SOME sense. For the moment - I don't want this to be a long post, quite apart from the fact that I'd rather not be thinking about any of this stuff at all! - I'll just note that one implication is that the biproduct defines a functor TP --> P, for any preadditive category P. (Of course, there are other fiddly details to verify as well, but this one will do for now.) Obviously the effect on objects of TP is the expected one: the object a = (a_i)_{i in I} maps to the biproduct (+)_i a_i, an object of P. As a minimum, I should check that an arrow (f_{i,j}: a_i --> b_j)_{(i,j) in IxJ} in TP maps (in a natural-looking way - formal naturalness is to be checked later, of course), to an arrow (+)_{i,j} f_{i,j}: a --> b in P. The (i)th row of the matrix f consists of the tuple of arrows f_{i,j} for all j in J. (Apologies again, in case* all my matrices have been transposed from the way they should be, and this row should really be a column. - Another reason why I shouldn't go too far with this now!) The biproduct (in what I vaguely recall is the usual way for a product) determines from these an arrow (x)_j f_{i,j}: a_i --> b; and then, in the usual way for a coproduct, determines from this tuple of arrows a_i --> b the arrow (+)_i ((x)_j) f_{i,j}): a --> b in P. Or, we can do it the other way around, obtaining instead the arrow (x)_j ((+)_i) f_{i,j}). *(In fact, I've definitely buggered up the ordering, because - I don't as a rule think about category theory at all! - all along I've been taking composition as a function hom(A, B) x hom(B, C) --> hom(A, C), (f, g) |--> f o g --- which is the wrong bloody way around! Sorry! I'll press ahead in the same misguided direction for the moment, but it all needs to be rewritten - unless somebody gives me a reference to where it's all been written out properly, already.) So the assumption here is that, for such a family of arrows f_{i,j}: a_i --> b_j, in a preadditive category P: (+)_i ((x)_j) f_{i,j}) = (x)_j ((+)_i) f_{i,j}) ... (*) I guess the easiest way to prove this is to argue that if two arrows g, h: a --> b in P are such that u_i o g o v_j = u_i o h o v_j for all i in I, and all j in J, where u_i: a_i --> a is the (i)th injection of the sum a = (+)_i a_i, and v_j: b --> b_j is the (j)th projection of the (bi)product b = (x)_j b_j, then g = h. Call this result (**). If (**) is true, then it does the trick, because, in (*) above: u_i o (RHS o v_j) = u_i o ((+)_i) f_{i,j}) = f_{i,j} (u_i o LHS) o v_j = ((x)_j) f_{i,j}) o v_j = f_{i,j} (**) is actually trivial (it's hard to see the wood for the trees!): because from (u_i o g) o v_j = (u_i o h) o v_j, for all j in J, it follows that u_i o g = u_i o h; and from this, for all i in I, it follows that g = h, as required. (Duh!) (I'm sure that: (a) all this is a standard result; (b) I've laboured and obfuscated the proof unnecessarily; (c) ... I forget now what (c) was, because it's past my bedtime ... I think it might have been that (*) and (**) should really have been proved together, as one result ... Anyway, this is another much-needed apology.) (Following on from this, I will later have to prove that the functor TP --> P is natural in P. Of course I won't do it now - as this is already a ridiculously long and fiddly post - but the next thing that does have to be done is quite similar in character:) >To do it properly, of course, one also has to define the action of T >on functors between preadditive categories. I expect this works. :-) Boring though it is, this really does have to be spelled out, to be sure that the whole thing isn't REAL abstract nonsense! Fortunately, it's not complicated: Let F: P --> Q be a functor between preadditive categories. We have to define a functor TF: TP --> TQ. For an object a = (a_i)_{i in I} in TP, define (TF)(a) = (F(a_i))_{i in I}. For an arrow f = (F_{i,j})_ {(i,j) in IxJ} in TP, define (TF)(f) = (F(f_{i,j}): F(a_i) --> F(b_j). It is clear that if f: a --> b in P, then (TF)(f): (TF)(a) --> (TF)(b) in TP. I think I'll cross my fingers and say it's obvious that TF is a functor. (I can always do penance later, but I must get to bed.) -- Angus Rodgers Contains mild peril === Subject: Re: Matrix elements? >When P is an additive category, this presumably relates in some nice >(but to me at the moment, horrible!) way to the biproduct considered >(assuming it can be!) as a natural transformation T --> I. [...] I'll just note that one implication is >that the biproduct defines a functor TP --> P, for any preadditive >category P. [...] Of course, I mean additive here, not preadditive. Now I really am about to turn the bedside light out, I promise! (Zzzzzzz ......) -- Angus Rodgers Contains mild peril === Subject: Re: Matrix elements? >Of course, the identity arrow on the object a = (a_i)_{i in I} is the >family of identity arrows 1(a) = (1(a_i): a_i --> a_i)_{i in I}. And of course, this is complete rubbish! Rather, we need to use identity maps and zero maps to construct an identity matrix. So I'll have to make sure that the categories we are using do indeed have zero maps to do this with. (But I am going to try very, very hard indeed not to think any more about any of this stuff tonight.) -- Angus Rodgers Contains mild peril === Subject: Re: Matrix elements? posting-account=a6woBRAAAADpNFZJBA7ZBx35zXaKmaP4 Gecko/2008060309 Firefox/3.0,gzip(gfe),gzip(gfe) i would like to know is there any definition what the matrix elements > could be? Of course, they can be numbers, alphanumerical signs, but > could they be also colors, symbols like dot, colon, semicolon? Is there any restriction what matrix elements could be? >If you just use a matrix as a table, then entries could be anything. >But as a *true* matrix, the entries should be elements of a set (or >class?) >where addition and multiplication is defined (a semiring). [...] > Reluctant though I am to get carried away with abstract nonsense, I > feel I must add that in order to give a coherent account even of such > very practical concepts as that of a partitioned matrix, it is useful > to allow matrix elements to belong to the hom-sets of an additive (or, > I suspect, even a preadditive) category: > æ æThus additive categories can be seen as the most general context > æin which the algebra of matrices makes sense. > (Perhaps one of our resident experts will take over from here, as I > don't at all trust myself with this stuff!) of matrices (over a ring) as an additive category, in >which the matrices are identified with morphisms. It >says little about entries of the matrices other than >assuming that these should belong to a (fixed) ring. Oh, dear! æI didn't read it closely, because I really am reluctant > to get into [what is in my hands] abstract nonsense. æI suppose > this obliges me to say precisely why I think it is useful to allow > matrix entries to be arrows in a preadditive category. æ(Additive > categories, I suppose, allow you to identify entries with elements > of a single hom-set using the biproduct, so that the generality isn't > needed.) I'll think about it in the bath - and perhaps decide after > all not to bother! æI'm quite sure there is a fearsomely elegant way > of doing it all - and equally sure that I'd mess it up, if I tried! I just don't want to have to think about Lawvere-style algebraic > theories, and how they might relate to all this. æBut the way in > which a partitioned matrix of partitioned matrices can be collapsed > into a partitioned matrix with entries of the same type does look > awfully like something in that theory ... or indeed something in > the theory of monads (which I've never got around to studying): I love/hate this kind of stuff, but I'm avoiding it like the plague, > for the next few years at least! æI did write some not-too-wild notes > on the topic a few decades ago, but destroyed them, along with all my > other notes on mathematics. I have a horrible feeling that I would end up writing something about > a natural transformation between two functors of the category of all > [small] preadditive categories to itself ... æ8-p it! æ:-) You see, if C is that category, then T: C -> C is a functor, where, > for any preadditive category, P, an object of TP is a finite family > (infinite matrices can also be considered in a less general context > ...) (a i) {i in I} of objects in P, and an arrow in TP: æf: (a i) {i in I} --> (b j) {j in J} is a finite family of arrows in P, (f {i,j}) {(i,j) in IxJ}, where: æf {i,j}: a i --> b j æin P (or it might be better to do it the other way around, I don't know). Composition of arrows in TP is defined by matrix multiplication: æ(f o g) {i,k} = sum {j in J} f {i,j} o g {j,k} æ[in hom(a i, c k)] where, of course, g: (b j) {j in J} --> (c k) {k in K} in TP. (Fog, indeed!) Without actually writing out the details, I am fairly sure there are > natural transformations I --> T and ToT --> T ... (etc., etc., etc.) When P is an additive category, this presumably relates in some nice > (but to me at the moment, horrible!) way to the biproduct considered > (assuming it can be!) as a natural transformation T --> I. To do it properly, of course, one also has to define the action of T > on functors between preadditive categories. æI expect this works. :-) Because I really am trying my absolute damnedest NOT to think about > it, all this is very likely to be wrong; but something vaguely like > it is probably right; and I'd rather leave the whole thing to the > experts - who have the taste and discretion to handle abstractions, > without going all baroque, or rococo, or plain loco (as I would do). two followups to the OP, deciding I'd better not get into the whole > thing at all; on the other hand, I am quite certain that there is a > perfectly level-headed approach to the topic of partitioned matrices > which makes modest use of a modicum of elementary category theory; > that's why I'd much rather leave the whole thing to someone else - > until I've studied mathematics for a few more years without trying > to think prematurely about any such abstract nonsense at all.) I thank you for taking the pains to explain your thought. The example of a partitioned matrix is illuminating. For example we might want to solve a system with structured entries: M = [ I A ] [ A' B ] If all these blocks had the same (square) dimensions, then we could view them as elements of a (possibly noncommutative) ring. But if the off-diagonal blocks A and A' are non-square (as is often the case), then we cannot look at things that way. Of course in a natural application we may hope to find that matrix operations are defined whenever we need them to be, and this is sort of the way we'd hope a categorical framework would assure matters work out. --c === Subject: Optimization of a Quasi-convex function posting-account=4VRU4AoAAAAZqfTm90VZhsCVGKpElpN0 Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) I have been trying to optimize a function that I know is quasi-convex using SeDuMi and YALMIP. However, the function does not converge to the global optimum and quite often gets stuck at two distinct values iterating between the two in consecutive iterations quite far from the global minima. This seems to be very unlikely of a quasi-convex problem for which a global minima is attainable. Does such a problem occur with a quasi-convex optimization problem? If so is there any workaround? -- Anil Nelakanti === Subject: Re: A Curious Question > As I said before, the term 'predicate' sounds impressive, but that's all > that's good about it. Its a pig of a word to use. Never use it again. > Well here you're just grasping at straws to excuse your own lack of > understanding of the purpose of the question. Try the same question another way: Is there anything not subject to 'not'? And the answer, of course, is no. > No thing is subject to not. It's self-evident. There is no state of not > a thing. > > Of course there is. Not A specifies everything not A. Everything is > subject to not and not is true of everything because not not is > self contradictory and hence false. Now I realize this is not obvious > to intuitionist mystics like philosophers, but it's necessarily true > because it tautologically exhausts all possibilities for truth. ~v~~ Everything that is not A or not A is not the same idea as A that is not. Also, the former does not describe A - 'not' is not a property of A. === Subject: Re: A Curious Question > I can say 'not is not predicated of a thing'. > You can say anything you want but that doesn't make it true. For > having said not is not predicated of a thing you've just predicated > not of a thing > 'not predicated of a thing' isn't a thing. Of course it is. What would make you think otherwise? The phrase not > predicated of a thing has properties which can be predicated of it. > The phrase can be talked about, described, and otherwise denoted. That conjures up a strange, weirdy blankness in my head. I don't want to muck about with the weight of ink or surface area of pixels in lieu of not is not predicated of a thing; so I suppose its not strange after all. > Even > the fact that you claim the phrase isn't a thing makes it a thing > because you're specifying analyzable predicates of the thing.. It's like the King of France then. === Subject: TI-89 vs. HP-50G for symbolic integrals posting-account=ZZvmuAkAAAAW2iAAgApuT91mi6Jectnm CLR 1.1.4322),gzip(gfe),gzip(gfe) Can someone comment on the reliability of either the TI-89 or the HP-50G for computing symbolic integrals. For study purposes I wish to compute integrals by hand and then check my work. This is what I would do with my graphing calculator when learning to graph functions. This worked very well, but my graphing calculator does not have integral or differentiation functions. The main point is to compute general indefinite symbolic integrals. For example, something like cos(x^2)/(3x^2 + 5x + 5). I want to confirm my on-paper integration operations are correct. I'm not trying to intentionally crash the calculator by pushing the limits of what can reasonably be expected from such a device (like raising something to the 1000th power and having fifteen embedded sub- functions). But it should unerringly compute the correct answers for reasonable integrals as you would find in a standard calculus textbook. === Subject: Riemann Sums I want to approximate the definite integral int_a^b f(x) dx by Riemann sums. What conditions must exist on f(x) to be able to do this? I have 1. f(x) is defined at a and b. 2. f(x) is (at least) piecewise continuous on [a, b]. Where am I going with this? I want to use Riemann sums to justify the statement int_a^{a+delta} f(x) dx = O(delta) for a general f(x) (subject to conditions). Naturally, I want the conditions to be as weak as possible. Can I do this? === Subject: Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > In order to prove that a Cauchy sequence with limit sqrt(2) in fact > has sqrt(2), you must show a number n_0 for eps = 10^-1000, such that > for all m and n > n_0 the absolute difference of a_m and a_n is less > than eps. False. The condition you mention is that of being a Cauchy sequence, > not that of converging to a particular value. The condition I use is that of being a limit of a Cauchy sequence. For every eps > 0 you have to give an n_0 such that for all m, n >= n_0 : | a_n - a_m| < eps. These a_m and a_n have to be taken from a sequence like a_n+1 = (a_n + 2/a_n)/2 with a_1 = 1, say. For eps = 10^-100 you cannot calculate a_m and a_n let alone all of them. But it is impossible to approximate sqrt(2) to better than > 10^-100. So you cannot find such a_m and a_n and, by the way, you > cannot find any approximation of sqrt(2) to better than 10^-100. This > idea is not a number. Given that the sequences you are talking about are not Cauchy > sequences at all, your claim of exhibiting a Cauchy sequence that does > not converge is wrong in your own terms. By the way you need not use Cauchy. The usual criterion is sufficient. There is no a_n that provably differs by less than 10^-100 from sqrt(2) because it is impossible to approximate sqrt(2) to 10^100 digits. === Subject: Re: A consideration concerning the diagonal argument of G. Cantor g2rfnb022rv@drn.newsguy.com> posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > On 12 Jun 2008 08:35:07 -0700, Daryl McCullough >stevendaryl3...@yahoo.com> said: Dik T. Winter says... >As the sentence is the largest natural smaller than 1, I do not know >how you come to your conclusion that modern set theorists would interprete >it as meaning 0. æAs far as I know, modern set theorists (except Bourbaki) >do not consider 0 as a natural number. I thought that set theorists typically identify natural numbers with > finite ordinals, which certainly includes 0. Of course, it is just a > matter of convention, but I thought that the convention was that 0 is > a natural number. I have never seen a text in which that wasn't so. You can be proud on your lack of education. Small wonder that you believe set theory to be the non plus ultra. === Subject: Re: Is the theory of topological vector spaces still alive? Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) Originator: israel@math.ubc.ca (Robert Israel) >Mathematicians do not acknowledge their duty to explain simply and >clearly why their field of interests is useful. Gradually they turn >into sportsmen whose aim is to impress the audience by their skill, >and nothing is important besides the skill. I agree that this kind of abuse is bad and should be fought, though I do not see it as the same thing as hyper-specialization. Even if a subject has connections to other fields, that doesn't automatically make it interesting and beautiful and important. You as a student may still not be convinced that it is a beautiful and important subject, and powerful experts may still abuse their power to force others to accept their point of view. It seems misguided to me to lay the blame on lack of connections to other fields per se, rather than on the abusive behavior of the people in question. More interesting to me is the question of whether it's a sign of bad health when a field becomes dominated by the construction of counterexamples. I think it certainly can be a bad sign, but even here I would be wary of over-generalization. Note that people usually don't complain if a subject contains lots of *examples*. But what makes something a counterexample rather than an example? It usually means that our naive intuitions about the subject are wrong. Some areas of mathematics may be particularly counterintuitive, so that we need an unusual number of (counter)examples to correct our intuition and map out the conceptual landscape properly. Devoting a period of time, even a rather long period of time, to building these counterexamples may in some cases be exactly the right thing to do. Of course, if the goal shifts from trying to understand natural mathematical structures to demonstrating one's virtuosity, then something has gone badly wrong. But again, this can happen in any field, specialized or not. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === Subject: Re: Is the theory of topological vector spaces still alive? posting-account=IsEI6goAAAB3ADiwRGmlND_myks93ylS Originator: israel@math.ubc.ca (Robert Israel) new one, because I again found mistakes there. Excuse me my English, and thank you in advance, Sergei Akbarov. ------------------------------------------------------------ Tim, I do not agree with you. > 1. ...Pick a random scholar in a random academic subject from a random > period > in history, and that scholar will assure you that the problem is > particularly > bad in that subject and at that period in time. Of course, what we discuss here happens very often in different parts of mathematics. But this does not mean that everywhere the situation is the same. For instance, Stephen mentioned here Banach spaces. Although, I am not impressed by what happens in the theory of Banach spaces, I nevertheless presume to prove that situation there is better than in topological vector spaces. difference between Banach and non-Banach situations is that in the first case the theory suggests a convenient class of objects that form a monoidal closed category (namely, class of Banach spaces) and for each monoid in this category (here monoids are nothing more nor less than Banach algebras) the corresponding modules over this monoid form enriched category over the initial category (of Banach spaces). In the theory of topological vector spaces the situation is absolutely different. For its lifetime this science did not create any class of spaces, convenient from the point of view of the customary algebraic intuition, i.e., a class that, like the class of Banach spaces, one could put into the place of the usual vector spaces in pure algebra. For those readers who are far from the category theory, this idea becomes clear after consideration the construction of algebra of endomorphisms. As is known, in pure algebra every module $X$ over an algebra $A$ generates an algebra $End_A(X)$ of endomorphisms of $X$ over $A$. This elementary fact ceases to be true in topological algebra, if we require algebras and modules to be complete (in some sense, general for all algebras and modules), and to have continuous multiplication (again in some sense, general for all these bjects). This can be conveniently illustrated by the following Exercise. Give a definition of topological algebra and topological module such that the following conditions hold: 1) all the topological modules are topological vector spaces and satisfy some standard condition of completeness (we need this to provide the convergence of natural nets and series); 2) the multiplication operations are continuous in some reasonable sense; 3) there is a natural procedure that endows the ring $End_A (X)$ of all endomorphisms of a given topological module $X$ over a given topological algebra $A$ with the structure of topological algebra with respect to your definition. You may be surprised, but up to the last time the only known solution of this Exercise in the frame of the theory of TVS was the class of Banach algebras and modules. (Note by the way that the appearance of Banach spaces is not a merit of the theory of TVS, since historically Banach spaces were studied before the general topological vector spaces. Moreover it was the narrowness of the class of Banach spaces that has lead to the appearance of the theory of TVS.) On hearing this one may ask: What were you, specialists in topological vector spaces, doing all this time? I asked them similar questions, and if we translate what they usually answer to the normal langauge, the translation will be as follows: Our counterexamples are more interesting for us. This is what we are proud of! So my first counter-argument is that the situation in different parts of mathematics is not the same, and we can compare it (here I agree with Stephen). And in this comparison the situation in the theory of TVS looks scandalous. > 2. It's not clear to me that it's a serious problem. Rather, it seems > to > me to be mostly harmless, a modest price to pay for the numerous good > papers > that get written. Moreover, I've suggested more than once already that > dormant may be a better word than dead. I do not agree with this as well. This is not very important, but, first, I want to say that I do not see numerous good papers that get written in the theory of TVS. And, second, I had not opportunity to make an experiment, but I am sure, if I replaced dead with dormant in my paper, the reaction would be the same: irritation. My main counter-argument here is as follows. If the idea that the hyper-specialization is a modest price for the progress becomes dominant in scientific society, and people imply from this that we should not be too exacting to what those hyper-specialists do, then we inevitably come to a situation when those hyper-specialists abuse their power. In practice this abuse looks as follows. When you are a student they tell you that this or that mathematical result is very important and elegant, and despite your doubts, you have to spend your time on studying innumerable counterexamples (which of course are the most convincing evidences of this beauty). As a corollary, by the time when you defend your PhD, you loose your human nature: you become a robot, who cannot distinguish useful things from useless things, beauty from deformity, and decency from dishonesty. Because questions like why is your science useful -- you treat as an invitation to bewilder the interlocutor by your professional skill. My reproach to mathematical society is that there is no culture here in such discussions. Mathematicians do not acknowledge their duty to explain simply and clearly why their field of interests is useful. Gradually they turn into sportsmen whose aim is to impress the audience by their skill, and nothing is important besides the skill. Those counterexamples by Enflo and others are indeed quite sophisticated. But if we treat them as progress in science, like people in TVS do, this becomes a speculation. Because in fact counterexamples are evidences of failure. That is my point. If you live in the West this problem, I suppose, is not of current importance for you, because you may have a lot of possibilities to change your company and to find like-minded persons, but when you live in a country like Russia, you become completely dependent on the opinion of those hyper-specialists, or perhaps we should say, skilful swindlers? :) So still I am curious if there are any specialist in the theory of topological vector spaces, who could explain these oddities in their science? === Subject: solution manual in PDF format in 24 hours posting-account=xYYqPAoAAABxyB7Wo6ji1HlF96uTfKBX Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) Hi! We organized group( GetSolution Team ) was founded in 2006 from the head of the filing to reduce the suffering students in the universities to find solutions to their books. please to know how to get solutions manual visit our website www.getsolutionteam.com we have a lot of solutions manual in low cast and in acrobat reader format PDF soft copy to get solution manual you want please send message to getsolution@hotmail.com getsolution(at)hotmail.com to see samples please visit our website www.getsolutionteam.com ............................................................................ ............................................................................ . ............. Mechanics of Fluids Solutions Manual by John Ward-Smith 8 ed Elements of Chemical Reaction Engineering By H Fogler, 3rd ed Advanced Engineering Mathematics by Erwin Kreyszig 8ed solutions manual Analysis and Design of Analog Integrated Circuits (4th Edition) Gray, Hurst, Lewis and Meyer Analytical Mechanics:1 Solutions Manual Grant R. Fowles, George L. Cassiday, 7 ed ANTENNAS FOR ALL APPLICATIONS, THIRD EDITION.txt solution Manual for C++ How to Program 3rd edition by deitel CMOS VLSI Design 3rd edition David Harris H E Weste Communication Systems 4Ed - A Bruce Carlson Solutions Manualt.txt communication systems engineering by poakis Computer Organization and Design, Revised Printing, 3rd Edition Solutions Manual By David A. Patterson, John L. Hennessy, Control Systems Engineering by Nise 4ed Device Electronics for Integrated Circuits Solutions Manual 3ed Digital Signal Processing - Proakis & Manolakis - Solutions Manual 3ed Discrete-Time Signal Processing 2nd Edition, 1999-02 by oppenheim Electronic Circuit Analysis and Design 2nd edt. by Donald A. Neamen Elementary Differential Equations and Boundary Value Problems , 8th Edition Elementary Principles of Chemical Processes Solutions Manual 3 ed By Richard M.Felder Engineering Circuit Analysis 6Ed - Hayt Solutions Manual.pdf Engineering Electromagnetics -Hayt (2001) Engineering Fluid Mechanics, 7th Edition - By Clayton T. Crowe, Donald Engineering Mechanics, Dynamics 5/e - Solutions manual By J. L. Meriam, L. G. Kraige, chapter 1-8 Fundamentals of Digital Logic with Verilog Design by s. Brown z vranesic Fundamentals of Logic Design 5th edition by charles roth Fundamentals of Machine Component Design 3ed - Solutions Manual By Robert C. Juvinall, Kurt Fundamentals of Physics, 7th Edition - Instructor's SOLUTIONS MANUAL halliday and resnick Solutions Manual to Econometric Analysis, 5th edition william h. Greene Heat Transfer: A Practical Approach. Solution Manual ONLY by cengel 2nd edition Introduction Fluid Mechanics, 6Th Edition Solution by fox introduction to algorithms 2nd edition instructors manual McGraw-Hill by thomas h. Cormen Introduction to VLSI Circuits and Systems (2001 draft) - John P Uyemura - Solutions Manual Introductory Quantum Optics by Knight and Gerry Jackson s Classical Electrodynamics 3rd edition James Stewart - Calculus 5Th Ed - Complete Instructor 's Solutions Manual 0534393217 linear systems and signals by lathi solutions manual blue covered Ljungqvist-Sargent.-.Recursive.Macroeconomic.Theory.-.Solutions.Manual. Mathematical Methods for Physics and Engineering A Comprehensive Guide 3rd edition by riley Microeconomic Theory - - Solutions Manual for Mas-Colell Modern Quantum Mechanics - J. J. Sakurai - Solution Physical Chemistry (Instructor's Solutions Manual) Peter Atkins & Julio de Paula 7ed Power System Analysis Solution Manual John Grainger, William D. Stevenson Principles of Electronic Materials and Devices, Solutions Manual ONLY Safa O. Kasap 2nd edition Probability and Statistics for Engineering and the Sciences [Solutions Manual] 6th edition by Jay L. Devore Roy D. Yates and David J. Goodman, Probability and Stochastic Processes - A Friendly Introduction for Electrical and Computer Engineers, 2nd edition, Recursive Methods in Economic Dynamics By Claudio Irigoyen ,Esteban RF circuit Design Theory and Application by Ludwig bretchko solution manuel Classical Mechanics, 2ed Partial Solutions Manual by Safko Semiconductor Device Fundamentals 1st edition by Robert F semiconductor physics and devices 3rd edition neamen Solutions Manual for the Guide to Energy Management 5 ed 2005-12 Klaus-Dieter E. Pawlik Solutions Manual to accompany Corporate Finance By Stephen A. Ross 6 edition Solutions of Engineering Electromagnetics -Hayt (2001).rar solutions Electric Machinery and Power System Fundamentals Thermodynamics: An Engineering Approach by: Yunus A. Cengel 5th edition Wireless Communications: Principles and Practice, 2nd edition theodore rappaport solutions manual modern control engineering by katsuhiko ogata 3rd edition ISBN: 0132273071 thomas calculus updated 10th edition instructor solution manual volume 1and 2 electric machinery fundamentals 4th edition stephen j chapman Problems and Solutions on Electromagnetism by zhao shu ping - you jun han Signal Processing and Linear Systems by lathi Hydraulics in Civil and Environmental Engineering By Andrew Chadwick Kc's Problems and Solutions for Microelectronic Circuits by k.c smith 4th edition principles of communication 5th edition ziemer william h. tranter solutions manual Linear algebra with applications 3rd edition Otto Bretcher - Solutions Manual Solution Manual: Probability and Random Processes for Electrical Engineering by Alberto Leon-Garcia not complete hw solutions Engineering Electromagnetics, 7th Ed., Hayt, Buck Linear algebra and it^s applications 3rd edition by david c. Lay Analytical Mechanics: Solutions Manual 7ed Grant R. Fowles, George L. Cassiday computer networks Andrew S. Tanenbaum 4th edition Design with Operational Amplifiers and Analog Integrated Circuits, 3rd edt. by F Instructor's Manual: Im Experiments with Economic Principles By Bergstrom Mechanics of Fluids Solutions Manual 8ed By John Ward-Smith Statistical Digital Signal Processing and Modeling - SOLUTIONS MANUAL By Monson H. Hayes Advanced Engineering Mathematics, 9th Edition By Erwin Kreyszig Solution to Advanced Macroeconomics 1996 romer Advanced Macroeconomics, Solutions Manual 1996 Romer Advanced Modern Engineering Mathematics, 3rd Edt by Glyn James - solution manuel APPLIED MATHEMATICS AND MODELLING FOR CHEMICAL ENGINEERS-8th Edition (solution manual) By Erwin Kreyszig T. Thornton, Jerry B. Marion Computational Techniques for Fluid Dynamics: A Solutions Manual By Karkenahalli Srinivas, Clive A. J. Fletcher DIGITAL DESIGN FOURTH EDITION by M. MORRIS MANO Mechanics of Materials, 6th Edition by James M. Gere (missing small portion, section 8.5) Essentials of Fluid Mechanics: Fundamentals and Applications, 1st Ed., by Cengel & Cimbala ISBN-13 9780073301129 Electric Circuits, Nillson Riedel , 8th edition Electric Circuits, Nillson Riedel , 7th edition Elementary Differential Equations And Boundary Value Problems, 7Th Ed - Boyce And Diprima Student Solutions Manual, Charles W Haines Ode Architect Companion Corporate Finance-7th Edition by Stephen A. Ross , Randolph W. Westerfield , Jeffrey Jaffe DATABASE MANAGEMENT SYSTEMS 3rd Edition by Ramakrishnan, Gehrke, Derstad, Seliko, Zhu- Solution Manual only odd solutions Design and Analysis of Experiments Solutions Manual 6th edition Digital and Analog Communication Systems - SOLUTIONS MANUAL By Leon W. Couch, Leon W. 7th edition student solutions Digital Integrated Circuits by Rabaey 2nd edition solution manuel chapter 3,5,6,10 Econometric Analysis [ONLY the Solutions Manual to the 6th Edition By William H. Greene instructor s resource manual to accompany electronic devices,6th edition and electronic devices electron flow version 4th edition thomas floyd Engineering Economy - Leland Blank & Anthony Tarquin 6th Edition selected solutions ( student solution) Engineering Fluid Mechanics, 7th Edition - Student Solutions Manual by Clayton T. Crowe, Donald F. Elger, John A. Roberson Engineering Mathematics, 4th edt. by John Bird - solution manual Finite Element Method: Volume 1 The Basis 5th edition by O. C. Zienkiewicz, R. L. Taylor Fundamentals of Chemical Reaction Engineering - Solutions Manual By Mark E. E. Davis, Robert J. J. Davis, Fundamentals of Electric Circuits 2nd by Alexander Sadiku Fundamentals of Engineering Electromagnetics--Cheng Fundamentals of Organic Chemistry, 5E, Study Guide and Solutions Manual By T. W. Graham Solomons Fundamentals of Probability With stochastic processes 3/e (Solutions Manual ) By Saeed Ghahramani Fundamentals of Thermal-Fluid Sciences Solution Manual 2ed By Yunus A. Cengel, Robert H. Turner, Fundamentals of Thermodynamics SOLUTION MANUAL 6ed By Richard E. Sonntag, Claus Borgnakke, Gordon J. Van Wylen, chemical Engineering: Solutions for Volumes 2 and 3 by coulson 2002-12-11 In Experiments with Economic Principles, Instructor's Manual By Bergstrom Instructor Manual to An Introduction to Thermodynamics and Statistical Mechanics 2ed By Keith Stowe Instructor Manual to Introduction to Solid State Physics Eighth Edition By Charles Kittel Instructor Manual to Introductory Quantum Optics By Christopher Gerry and Peter Knight Instructor Manual to SEMICONDUCTOR DEVICES Physics and Technology Second Edition By S.M.Sze Instructor Solutions Manual to accompany Boyce Elementary Differential Equations and Boundary Value Problem 8ed by Charles W. Haines, William E. Boyce Instructor's Manual for Solving ODEs with MATLAB By L. F. Shampine, I. Gladwell, S. Thompson Instructor's Solutions for: Design of Analog CMOS Integrated Circuits by razavi Advanced Engineering Mathematics 3rd Edition by Dennis G Zill and Michael R Cullen Wireless Communication and Networks second edition William Stallings solutions manual Transport Phenomena - 2nd edition by Bird, Stewart and Lightfoot Solution Manual Signals and Systems 2nd edition simon Haykin Solutions Manual Electrical Engineering: Principles and Applications 3ed Allan R. Hambley Introduction to Chemical Engineering Thermodynamics 7th edition (solution manual) By J.M. Smith, Hendrick C Van Ness Introduction to Heat Transfer - 013391061X Solution's Manual By peyman pourmoghaddam , Vedat S. Arpaci not complete Computational Techniques for Fluid Dynamics: A Solutions Manual By Karkenahalli Srinivas, Clive A. J. Fletcher Introduction to Heat Transfer 4th Edition SOLUTION MANUAL By Frank P. Incropera, David P. DeWitt Heat Transfer-Fundamentals of Heat and Mass Transfer-Incropera & Dewitt Solution Manual power analys and design by glover 3rd ediiton Digital Communications: Fundamentals And Applications (2nd Edition)- Bernard Sklar Engineering mechanics: Dynamics - 4th Ed. by Bedford and Fowler design of machinary by norton 3rd edition Digital Signal Processing (2nd Ed.) (Mitra) Solution Manual Electronic Circuit Analysis and Design 2nd edt. by Donald A. Neamen - solution manuel Introduction to Probability by Dimitri P. Bertsekas - solution isbn: 1-886529-40-X INTRODUCTION to PROBABILITY Charles M. Grinstead and J. Laurie Snell computer networking a top down approach 3rd edition solution manual by James F.Kurose, Keith W. Ross Solution manual of sedra microelectronic circuits 5th ed. By Adel S. Sedra, Kenneth C. Smith Elementary Differential Equations And Boundary Value Problems, 7Th Ed - Boyce And Diprima Student Solutions Manual, Charles W Haines Ode Architect Companion Fundamentals of Electronic Circuit Design by David and Donal Comer Introduction to Quantum Mechanics (Second Edition) - Solutions Manual By David J. Griffiths Introductory Econometrics A Modern Approach 2ed Jeffrey Wooldridge Modern Digital and Analog Communications Systems - B P Lathi Solutions Manual 3rd edition Solutions Manual to accompany Investment Analysis & Portfolio Management, 7th edition by Reilly and Brown Selected Answers-Basic Engineering Circuit Analysis-7th Ed. by J. David Irwin Shigley's Mechanical Engineering Design 8th Ed - Solution Manual By by Richard Budynas, J. Keith Nisbett Shigley's Mechanical Engineering Design 7th Ed - Solution Manual By by Richard Budynas, J.Charles Mischke Soil Mechanics Solutions Manual (2nd Edition) By William Powrie Solution Manual for Numerical Solution of Partial Differential Equations: An Introduction 2ed By K. W. Morton, D. F. Mayers Solution Manual-Microelectronics-Digital and Analog Circuits and Systems by Millman Solutions Manual for Materials Science and Engineering An Introduction 6E By William D. Callister Solutions Manual for System Dynamics 3rd Ed By Katsuhiko Ogata Statistical Digital Signal Processing and Modeling - SOLUTIONS MANUAL By Monson H. Hayes Engineering Mechanics, statics 5th edition Solutions manual By J. L. Meriam, L. G. Kraige chapter 1-7 Probability and Statistics for Engineers and Scientists Manual HAYLER Solutions Manual Physics for Scientists and Engineers Extended Version, 5th edition, PROCESS SYSTEMS ANALYSIS AND CONTROL - DONALD R. COUGHANOWR Solution Manual Ulaby Applied Electromagnetics Butterworth Heinemann - Coulson And Richardson - Chemical Engineering Vol I (Solutions Manual V Edition) Instructor's Solutions Manual for Serway and Jewett's Physics for Scientists and Engineers Sixth Edition Volume One by Ralph V. McGrew Instructor's Solutions Manual for Serway and Jewett's Physics for Scientists and Engineers Sixth Edition Volume two by Ralph V. McGrew Digital Communications, 4th edition, 2000-08 book+solution by John Proakis Fundamentals of Wireless Communication by Tse and Viswanath Probability & Statistics for Engineers & Scientists, 8th Edition: Instructors Solution Manual ONLY by Sharon Myers , Keying Ye, Walpole Undergraduate Econometrics Solutions Manual - Hill, Judge and Griffiths CMOS Analog Circuit Design, 2ed Solutions by Phillip E. Allen, Douglas R. Holberg Solutions Manual For An Introduction To The Finite Element Method, 3rd Edition by J. N. Reddy Engineering Circuit Analysis 7ed by Hayt ISBN-10: 0071109374 ( please ask before sell or send sample) be carefullllllll Harcourt Mathematics 12 Geometry and Discrete Mathematics Solutions Manual By McGraw-Hill solutions manual for Mechanics of Materials Sixth Edition by R.C.Hibbeler by F. P. Beer, E. R. Johnston Jr., E. R. Eisenberg, & G. H. Staab Solutions Manual Vector Mechanics for EngineersSTATICS, 7th Edition by F. P. Beer, E. R. Johnston Jr., E. R. Eisenberg, & G. H. Staab Calculus Single Variable 4ed chapter 1 to 11 Hughes-Hallett, Gleason, McCallum, et al. Fourier and Laplace Transform - Antwoorden Field and Wave Electromagnetics, 2nd edition, Cheng Thomas Calculus 11th edition Multivariable Calculus: Student Solutions Manual 4th edition by James Stewart (Author) solutions manual for Introduction to Electrodynamics (Third Edition) by David J. Griffiths Engineering Mathematics, 4th Edition, Solution Manual by NEWNES APPLIED MATHEMATICS AND MODELLING FOR CHEMICAL ENGINEERS-8th Edition (solution manual) by Erwin Kreyszig Electric Machinery and Power System Fundamentals Stephen J. Chapman first edition Chemical Engineering Design, Fourth Edition: Chemical Engineering Volume 6 (Coulson & Richardson's Chemical Engineering) Student Solutions Manual and Study Guide to accompany Fundamentals of Fluid Mechanics, 5th Edition by Bruce R. Munson Donald F. Young, Theodore HARCOURT MATHEMATICS 12 Advanced Functions and Introductory Calculus - Solutions Manual by Brian / Nelson Lagrangian and Hamiltonian Mechanics Solutions to the Exercises by M. G. Calkin Linear algebra with applications 3e Otto Bretcher - Solutions Manual Soil Mechanics Concepts and Applications By William Powrie Fundamentals of Financial Management, 12th Edition (Instructors guide ONLY) by James C. Van Horne, John M Wachowic An Introduction to Numerical Analysis by Endre S.9fli, David F. Mayers Feedback Control of Dynamic Systems 4th edition Franklin - Solutions Manual Solution Manual : Nanoengineering of Structural, Functional and Smart Materials Solution Manuals +book for The Economics of Financial Markets By Roy E. Bailey Instructor's Manual Im Experiments with Economic Principles By Bergstrom Instructor Manual to A Course in Modern Mathematical Physics By Peter Szekeres Fundamentals of Chemical Reaction Engineering - Solutions Manual By Mark E. E. Davis, Robert J. J. Davis Solutions Manual for Equilibrium and Non-Equilibrium Statistical Thermodynamics By Michel Le Bellac Introduction To Electric Circuits 6th Ed [Solutions Manual] By R. C. Dorf and J. A. Svoboda Chemical Engineering Solutions manual for Volumes( 2 and 3) 3 edition Backurst J. R., Harker J.H. & Richardson J. F Fluid Mechanics with Student Resources, 5th edition 2002-12 by Frank M. White Instructor Manual to Quantum Physics Third Edition by Stephen Gasiorowicz Solutions Manual for use with Fundamentals of Corporate Finance, 4th Edition (Brealey, Myers, Marcus) by Bruce Swenson Investment Analysis and Portfolio Management- Solutions Manual, 7th Edition by Frank K. et al. Reilly Microeconomic Analysis Solution Manual - Varian 3rd edition Structural Analysis by Hibbeler 5th edition The Science and Engineering of Materials by Donald R. Askeland Frank Haddleton 4th edition solutions manual for Engineering Mechanics - Statics (11th ) by R.C.HIBBELER Differential Equations & Linear Algebra, 2nd ed., Farlow modern electronic communication 8th edition by gary miller and beasley Instructor's Manual for Solving ODEs with MATLAB By L. F. Shampine, I. Gladwell, S. Thompson Solution Manual analysis design of analog IC design Microwave And Rf Design Of Wireless Systems - Solution Manual (D M Pozar) Solution Manual Peterson and DavieÍs Computer Networks: A Systems Approach 2nd edition Solution Manual Communication Networks Fundamental concepts & key Architectures By Leon Garcia Widjaja not complete 3 4 5 6 7 8 10 Solution Manual - Electrical Machines, Drives and Power Systems 6th edition ISBN 0131969188 Logic Design and Digital Fundamentals - Moris Mano partial solutions Probability, Random Variables, and Stochastic Processes [Only Solutions Manual] by Athanasios Papoulis ,S.Unnikrishna Pillai 4th edition ntroduction to Environmental Engineering and Science (2nd Edition) (Hardcover) by Gilbert M. Masters Classical Mechanics (2nd Edition) by Herbert Goldstein Data & Computer Communications, 7th Edition [Solutions Manual only] by William Stallings Calculus Third Editon By Strauss, Bradley and Smith Fundamentals of Semiconductor Devices - Anderson SILBERSCHATZ , PETER BAER GALVIN, GREG GAGNE Electrical Machines Drives and Power Systems 6th edition by Theodore Wildi Modern Control System 9 Solution Drof Bishop Electric Machinery by A. E. Charles Kingsley, Jr.Fitzgerald 6th edition 0073660094 A First Course In Probability Solution Manual,Ross 6th Fundamentals of Electromagnetics with Engineering Applications by Stuart M. Wentworth Data Communications and Networking fourth edition by Behrouz A.Forouzan Signals and Systems, Prentice-Hall Oppenheim, Willsky, Young 2nd edition The Science and Engineering of Materials by Donald R. Askeland Frank Haddleton 4th edition FESTO DIDACTIC Process Control System Microwave Engineering 3e - David M Pozar A First Course in Abstract Algebra 7Ed - Solutions Manual by Fraleigh Addison Wesley - Distributed Systems, Concepts and Design (Exercise Solutions) - G. Coulouris, J. Dollimore and T. Kindberg allen holberg cmos analog circuit design 2nd edition homework solutions An introduction to the mathematics of financial derivatives Neftci solution manual Andrew Tanenbaum Structured Computer OrganizationSolutions Manual Bioprocess Engineering Principles - Solutions Manual (Original) by pauline m. Doran Carey, Study guide and solution manual for organic chemistry Chapra Applied Numerical Methods With Matlab For Engieers Solutions Manual Chiang & Wainwright 2005 Instructor's Manual Of Fundamental Methods Of Mathematical Economics Mc Graw Hill Computer Networks Systems Approach3ed by davie peterson solutions manual Digital image processing - Gonzalez 2Ed- Solutions Manual (209p) Doets & Eijck - The Haskell Road To Logic, Math and Programming - Solutions to Exercises Econometrics - [Instructor Solution Manual] The Econometrics of Financial Markets john y. campbell, andrew w. Lo Elements Of Information Theory - Solution Manual by thomas m. cover and joy a. Thomas Fundamentals of Actuarial Mathematicsî by D. Promislow Instructor's Manual for Sipser's Introduction to the Theory of Computation by Ching Law Instructor's Manual(Information Technology Project Management 3Rd Edition) by Kathy Schwalbe Instructor's solution manual ISBN 0534382150 A Transition to Advanced Mathematics solution manual by douglas smith 5th edition Introduction To Wireless Systems - P M Shankar - Solutions Manual Mathematics for Economists Solution Manual - Simon and Blume (ver 2) Microeconomic Analysis Third Edition by Hal R. Varian Operating System Concepts (6Th Ed)-Instructor'S Manual (A Silberschatz) Operating System Concepts, 7th Edtion, Instructor's Manual by A Silberschatz Papathomas T.V. - Solutions Manual to Accompany Millman Integrated Electronics Analog and Digital Circuits and Systems (McGraw-Hill) Pattern Recognition and Machine Learning (Solution Manual) - Bishop Solutions Manual for Statistical Inference - Casella & Berger Solutions Manual Nanoengineering of Structural Functional and Smart Materials Solutions-Mathematics for Economists Solution Manual (Blume, 1994) STRUCTURED COMPUTER ORGANIZATION 5ed ANDREW S. TANENBAUM Solution Manuals for Advanced Dynamics by Donald T. Greenwood Solution Manuals for The Economics of Financial Markets by Roy E. Bailey [Math] Haberman Applied Partial Differential Equations 4e Instructor's Manual A Quantum Approach to Condensed Matter Physics Solutions by philip l. Taylor An Introduction to Economic Dynamics 0521804787 sols Engineering Mechanics Dynamics 3rd edition solution manual Hibbeler R.C. updated fixed 09-2006 Fundamentals of Quantum Mechanics 0521829526 sols Organic Chemistry 2nd ed [Student SOLUTIONS MANUAL and Study Gde] - J. Hornback, B. Murugaverl (Thomson, 2006) WW RF Circuit Design Theory and Applications Solution Manual by reinhold ludwig pavel bretchko Solution Manual for A First Course in String Theory chapter 1 to 16 Solution manual Thermodynamics Of TurboMachinery 5th edition Solution Manual for Mathematical Models in Biology An Introduction by Elizabeth S. Allman john a. Rhodes Solution Manual for Classical Dynamics A Contemporary Approach by Jorge V. Jose, Eugene J. Saletan Solution Manual for An Introduction to Ordinary Differential Equations laser fundamentals 2nd edition by william t. Silfvast Elements of Electromagnetics, 3rd Ed., Matthew N.O. Sadiku A FIRST COURSE IN DIFFERENTIAL EQUATIONS WITH MODELING APPLICATION 7TH EDITION BY DENNIS G. ZILL Solution Manual Microelectronics Circuit Analysis Desing 3rd Edition by R. Jaeger Wiley Chemical And Engineering Thermodynamics 3Ed Solutions Manual Fracture mechanics fundamentals and applications 2nd edition anderson solution manual modern control engineering by katsuhiko ogata 4th edition Thermodynamics an engineering approach sixth edition ( SI units ) : solutions manual by Yunus A. Cengel, Michael A. Boles Fluid Mechanics - Solutions Manual by CENGEL A Guide to Physics Problems, Part 1 - Mechanics, Relativity, and Electrodynamics A Guide to Physics Problems, Part 2 - Thermodynamics, Statistical Physics, and Quantum Mechanics linear algebra Juan de Burgos Solutions Manual spanish Applied Statistics and Probability for Engineers Student 3rd.Ed Classical Electrodynamics - 2nd Ed. John David Jackson byKasper van Wijk Chemical and Engineering Thermodynamics- 3rd Edition- Solutions Manual Communication Systems Engineering Proakis J (2002) Solutions Manual Daniel Shanks - Solved And Unsolved Problems In Number Theory (2Nd Ed), 1978 Design of Analog CMOS Integrated Circuits [McGraw Hill Solutions Manual Digital Signal Processing A Computer-Based Approach 1st ed Solutions Manual mitra Elementary Mechanics & Thermodynamics [2000 by Professor Jhon W. Norbury Engineering - Materials Science, Milton Ohring Solutions Manual Fundamentals Of Fluid Mechanics 3Rd And 4Th Edition Solutions Manual Fundamentals of Thermodynamics [Sonntag-Borgnakke-Van Wylen Solutions Manual volume 1 and volume 2 Introduction to Fluid Mechanics (Fox, 5th ed) Solutions Manual Juvinall, Marshek - Fundamentals of Machine Component Design, 3rd ed - Student Solutions Manual McgrawHill - William H. Hayt, John A. Buck - Engineering Electromagnetics, 6ed Solutions Manual Physics by Resnick Halliday Krane, 5th Ed. Vol 2 Schaums Mathematical Handbook of Formulas and Tables Solution To Two-Dimensional Incompressible Navier-Stokes Equations- Maciej Matyka Prentice Hall - Solutions Manual; Communication Systems Engineering University Physics - Sears and Zemansky's 11th Ed Solutions Manual Wankat & Oreovicz - Teaching Engineering Applied Strength of Materials (4th Edition) SOLUTION MANUAL by Robert L. Mott University physics 11th edition solution manual by Young and Freedman Solution Automatic Control Systems 8Ed by Kuo and Golnaraghi Solution Manuals for Fundamentals of Quantum Mechanics For Solid State Electronics and Optics by C. L. Tang Solution Manuals for Applied Quantum Mechanics by A. F. J. Levi Digital Design (3rd Edition) by M. Morris Mano High-Speed Digital System Design A Handbook of Interconnect Theory and Design Practicesbby Stephen H. Hall IBM WebSphere RFID Handbook A Solution Guide by IBM Redbooks Solution Manual for An Introduction to Numerical Analysis by Endre Suli Solution Manual for Classical Mechanics by R. Douglas Gregory Engineering Mechanics Dynamics (11th Edition) by Russell C. Hibbeler Adaptive Control 2nd. Edt. by Karl.J.Astrom - solution manuel Applied Numerical Analysis 7Ed - Curtis F. Gerald, Patrick O. Wheatley - Solutions manual Digital Signal Processing by Thomas J. Cavicchi - solution manuel Discrete Time Signal Processing 2nd Ed - Solutions to Exercises Linear circuit analysis by R. A. DeCarlo and P. Lin - solution manuel principles and application of elect eng giorigo rizzoni Solid State Electronic Device by Ben Streetman Solution Manual Device Electronics for Integrated Circuits 3Edition Muller Kamins Solution Manual Electronic Physics Strabman Solution to Skill - Assessment Exercises to Accompany Control Systems Engineering 3rd edt. by Norman S. Nise Mechanics Of Materials Solution Manual (3Rd Ed , By Beer) nearly same 4th edition just numbered different materials science and engineering,callister 7th edition (password aguara) Computer Organization and Design: The Hardware/Software Interface, 3rd Edition by David A. Patterson, John L. Hennessy Complete solutions manual for Zill s First Course in Differential Equations with Modeling Applications (7th ed.) and Zill & amp; Cullen s Diferential Equations with Boundary-Value Problems (5th ed.) Introduction To Linear Algebra 3Ed - Gilbert Strang Solutions Manual Physics for Scientist & Engineers with Modern Physics - A strategic Approach chapter by Randall D. Knight 1-35 Signal Processing First-Mclellan, Schafer & Yoder Solution Manual chapter 3-12 Physics Principles with Applications Instructor's Solutions Manual (Giancoli, Volume 1and 2) 6th edition Anton Bivens Davis CALCULUS early transcendentals 7th edition Mechanical Vibrations - Singiresu Rao - Solutions Manual 3rd edition chapters missing 6 9 12 ( all links same ) 35 mb Numerical Methods For Engineers Solutions Manual by chapra 4th edition Computer Architecture Pipelined and Parallel Processor Design (Solutions Manual) by micheal j. flynn selected solutions Mathematical Models in Biology Sols by Elizabeth S. Allman, John A. Rhodes Dynamics of Mechanical Systems Solutions Manual (Horwood Engineering Science Series) by C. T. F. Ross Instructor Manual to Special Relativity by Patricia M. Schwarz and John H. Schwarz Solution Manuals for An Introduction to Economic Dynamics by Ronald Shone Quantum Field Theory (draft version) & Instructor's Manual by Mark Srednicki book + solution manual Solution manual for Operating systems Internals and Design principles 4th by william stallings Engineering Mechanics Statics 11th Edition By R.C.Hibbeler Solutions manual for Fundamentals of engineering thermodynamics moran shapiro Solutions Manual to accompany Corporate Finance By Stephen A. Ross 8th edition Advanced Digital Design with the Verilog HDL Michael D. Ciletti selected solutions Engineering Mechanics: Dynamics 2 Ed. by Riley and Sturges Kinetics of Catalytic Reactions--Solutions Manualby: M. Albert Vannice Solutions Manual for Techniques of Problem Solving by luiz fernandez Unit Operations of Chemical Engineering, 7th Edition, Solutions Manual by: Warren McCabe, Julian Smith, Peter Harriott Solution Manual (complete) Communication Networks Fundamental concepts & key Architectures Alberto Leon-Garcia Financial Accounting 6th edition by Harrison Solution Manual by harrison Solution Manual-Fundamentals of Digital Logic with VHDL Design-1st edition by S. Brown, Z. Vranesic Mechanics of Materials [solutions manual] hibbeler 7th edition instructor s manual with powerpoints to accompany pic microcontroller and embedded systems by muhammed ali mazidi rolin d mckinlay Accounting Concepts and Applications (9th Ed.) by W. Steve Solutions Manual to Accompany Organic Chemistry by Robert C. Atkins, Francis A Carey, Robert Atkins, Francis Carey solution manual differential equations 5th edition by zill Fundamentals of Electric Circuits 3rd edition by Alexander Sadiku Advanced Engineering Mathematics Dennis G Zill 2nd Solution solution manual Principles of corporate finance 7e Instructors Solutions Manual for Differential Equations with Boundary Value Problems, 2/E by john polking Solution Manual for Basic Engineering Circuit Analysis, 8th Edition by J. David Irwin, R. Mark Nelms Solution manual for Calculus A Complete Course 6th by R.A. Adams Elementary Linear Algebra with Applications 9 edition by Howard Anton, Chris Rorres student manual Elementary Linear Algebra with Applications 9 edition by Howard Anton, Chris Rorres Instructor Solutions Manual and Instructor Testbank Fundamentals of Engineering Thermodynamics: Si Version / 5th Edition by Michael J. Moran Howard N. Shapiro === Subject: Re: Jacobi Triple Product <22658297.1213599753957.JavaMail.jakarta@nitrogen.mathforum.org>, > I need help concearning Jacobi Triple Product. I don't understand the proof. > Can anyone help me, please? > Raul What is the theorem whose proof you do not understand? I seem to recall more than one theorem about the Jacobi triple product. === Subject: Re: Jacobi Triple Product I don't know how to give links, so I'll just type 3 adresses. How can I give links? http://www.math.uwaterloo.ca/~dmjackso/CO630/JTPID.pdf It's Theorem 1.2. http://en.wikipedia.org/wiki/Jacobi_triple_product http://mathworld.wolfram.com/JacobiTripleProduct.html I don't understand any of the three. === Subject: Re: Jacobi Triple Product I just saw that it automatically gives links, I should of have previewed my post, I think, sorry. === Subject: solutions manual from GetSolution Team posting-account=-MgRFgoAAABvI7JSeOO7YLmYCcWsSE66 Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) Hi! We organized group( GetSolution Team ) was founded in 2006 from the head of the filing to reduce the suffering students in the universities to find solutions to their books. please to know how to get solutions manual visit our website www.getsolutionteam.com we have a lot of solutions manual in low cast and in acrobat reader format PDF soft copy to get solution manual you want please send message to getsolution@hotmail.com getsolution(at)hotmail.com to see samples please visit our website www.getsolutionteam.com ............................................................................ ............................................................................ . ............. Mechanics of Fluids Solutions Manual by John Ward-Smith 8 ed Elements of Chemical Reaction Engineering By H Fogler, 3rd ed Advanced Engineering Mathematics by Erwin Kreyszig 8ed solutions manual Analysis and Design of Analog Integrated Circuits (4th Edition) Gray, Hurst, Lewis and Meyer Analytical Mechanics:1 Solutions Manual Grant R. Fowles, George L. Cassiday, 7 ed ANTENNAS FOR ALL APPLICATIONS, THIRD EDITION.txt solution Manual for C++ How to Program 3rd edition by deitel CMOS VLSI Design 3rd edition David Harris H E Weste Communication Systems 4Ed - A Bruce Carlson Solutions Manualt.txt communication systems engineering by poakis Computer Organization and Design, Revised Printing, 3rd Edition Solutions Manual By David A. Patterson, John L. Hennessy, Control Systems Engineering by Nise 4ed Device Electronics for Integrated Circuits Solutions Manual 3ed Digital Signal Processing - Proakis & Manolakis - Solutions Manual 3ed Discrete-Time Signal Processing 2nd Edition, 1999-02 by oppenheim Electronic Circuit Analysis and Design 2nd edt. by Donald A. Neamen Elementary Differential Equations and Boundary Value Problems , 8th Edition Elementary Principles of Chemical Processes Solutions Manual 3 ed By Richard M.Felder Engineering Circuit Analysis 6Ed - Hayt Solutions Manual.pdf Engineering Electromagnetics -Hayt (2001) Engineering Fluid Mechanics, 7th Edition - By Clayton T. Crowe, Donald Engineering Mechanics, Dynamics 5/e - Solutions manual By J. L. Meriam, L. G. Kraige, chapter 1-8 Fundamentals of Digital Logic with Verilog Design by s. Brown z vranesic Fundamentals of Logic Design 5th edition by charles roth Fundamentals of Machine Component Design 3ed - Solutions Manual By Robert C. Juvinall, Kurt Fundamentals of Physics, 7th Edition - Instructor's SOLUTIONS MANUAL halliday and resnick Solutions Manual to Econometric Analysis, 5th edition william h. Greene Heat Transfer: A Practical Approach. Solution Manual ONLY by cengel 2nd edition Introduction Fluid Mechanics, 6Th Edition Solution by fox introduction to algorithms 2nd edition instructors manual McGraw-Hill by thomas h. Cormen Introduction to VLSI Circuits and Systems (2001 draft) - John P Uyemura - Solutions Manual Introductory Quantum Optics by Knight and Gerry Jackson s Classical Electrodynamics 3rd edition James Stewart - Calculus 5Th Ed - Complete Instructor 's Solutions Manual 0534393217 linear systems and signals by lathi solutions manual blue covered Ljungqvist-Sargent.-.Recursive.Macroeconomic.Theory.-.Solutions.Manual. Mathematical Methods for Physics and Engineering A Comprehensive Guide 3rd edition by riley Microeconomic Theory - - Solutions Manual for Mas-Colell Modern Quantum Mechanics - J. J. Sakurai - Solution Physical Chemistry (Instructor's Solutions Manual) Peter Atkins & Julio de Paula 7ed Power System Analysis Solution Manual John Grainger, William D. Stevenson Principles of Electronic Materials and Devices, Solutions Manual ONLY Safa O. Kasap 2nd edition Probability and Statistics for Engineering and the Sciences [Solutions Manual] 6th edition by Jay L. Devore Roy D. Yates and David J. Goodman, Probability and Stochastic Processes - A Friendly Introduction for Electrical and Computer Engineers, 2nd edition, Recursive Methods in Economic Dynamics By Claudio Irigoyen ,Esteban RF circuit Design Theory and Application by Ludwig bretchko solution manuel Classical Mechanics, 2ed Partial Solutions Manual by Safko Semiconductor Device Fundamentals 1st edition by Robert F semiconductor physics and devices 3rd edition neamen Solutions Manual for the Guide to Energy Management 5 ed 2005-12 Klaus-Dieter E. Pawlik Solutions Manual to accompany Corporate Finance By Stephen A. Ross 6 edition Solutions of Engineering Electromagnetics -Hayt (2001).rar solutions Electric Machinery and Power System Fundamentals Thermodynamics: An Engineering Approach by: Yunus A. Cengel 5th edition Wireless Communications: Principles and Practice, 2nd edition theodore rappaport solutions manual modern control engineering by katsuhiko ogata 3rd edition ISBN: 0132273071 thomas calculus updated 10th edition instructor solution manual volume 1and 2 electric machinery fundamentals 4th edition stephen j chapman Problems and Solutions on Electromagnetism by zhao shu ping - you jun han Signal Processing and Linear Systems by lathi Hydraulics in Civil and Environmental Engineering By Andrew Chadwick Kc's Problems and Solutions for Microelectronic Circuits by k.c smith 4th edition principles of communication 5th edition ziemer william h. tranter solutions manual Linear algebra with applications 3rd edition Otto Bretcher - Solutions Manual Solution Manual: Probability and Random Processes for Electrical Engineering by Alberto Leon-Garcia not complete hw solutions Engineering Electromagnetics, 7th Ed., Hayt, Buck Linear algebra and it^s applications 3rd edition by david c. Lay Analytical Mechanics: Solutions Manual 7ed Grant R. Fowles, George L. Cassiday computer networks Andrew S. Tanenbaum 4th edition Design with Operational Amplifiers and Analog Integrated Circuits, 3rd edt. by F Instructor's Manual: Im Experiments with Economic Principles By Bergstrom Mechanics of Fluids Solutions Manual 8ed By John Ward-Smith Statistical Digital Signal Processing and Modeling - SOLUTIONS MANUAL By Monson H. Hayes Advanced Engineering Mathematics, 9th Edition By Erwin Kreyszig Solution to Advanced Macroeconomics 1996 romer Advanced Macroeconomics, Solutions Manual 1996 Romer Advanced Modern Engineering Mathematics, 3rd Edt by Glyn James - solution manuel APPLIED MATHEMATICS AND MODELLING FOR CHEMICAL ENGINEERS-8th Edition (solution manual) By Erwin Kreyszig T. Thornton, Jerry B. Marion Computational Techniques for Fluid Dynamics: A Solutions Manual By Karkenahalli Srinivas, Clive A. J. Fletcher DIGITAL DESIGN FOURTH EDITION by M. MORRIS MANO Mechanics of Materials, 6th Edition by James M. Gere (missing small portion, section 8.5) Essentials of Fluid Mechanics: Fundamentals and Applications, 1st Ed., by Cengel & Cimbala ISBN-13 9780073301129 Electric Circuits, Nillson Riedel , 8th edition Electric Circuits, Nillson Riedel , 7th edition Elementary Differential Equations And Boundary Value Problems, 7Th Ed - Boyce And Diprima Student Solutions Manual, Charles W Haines Ode Architect Companion Corporate Finance-7th Edition by Stephen A. Ross , Randolph W. Westerfield , Jeffrey Jaffe DATABASE MANAGEMENT SYSTEMS 3rd Edition by Ramakrishnan, Gehrke, Derstad, Seliko, Zhu- Solution Manual only odd solutions Design and Analysis of Experiments Solutions Manual 6th edition Digital and Analog Communication Systems - SOLUTIONS MANUAL By Leon W. Couch, Leon W. 7th edition student solutions Digital Integrated Circuits by Rabaey 2nd edition solution manuel chapter 3,5,6,10 Econometric Analysis [ONLY the Solutions Manual to the 6th Edition By William H. Greene instructor s resource manual to accompany electronic devices,6th edition and electronic devices electron flow version 4th edition thomas floyd Engineering Economy - Leland Blank & Anthony Tarquin 6th Edition selected solutions ( student solution) Engineering Fluid Mechanics, 7th Edition - Student Solutions Manual by Clayton T. Crowe, Donald F. Elger, John A. Roberson Engineering Mathematics, 4th edt. by John Bird - solution manual Finite Element Method: Volume 1 The Basis 5th edition by O. C. Zienkiewicz, R. L. Taylor Fundamentals of Chemical Reaction Engineering - Solutions Manual By Mark E. E. Davis, Robert J. J. Davis, Fundamentals of Electric Circuits 2nd by Alexander Sadiku Fundamentals of Engineering Electromagnetics--Cheng Fundamentals of Organic Chemistry, 5E, Study Guide and Solutions Manual By T. W. Graham Solomons Fundamentals of Probability With stochastic processes 3/e (Solutions Manual ) By Saeed Ghahramani Fundamentals of Thermal-Fluid Sciences Solution Manual 2ed By Yunus A. Cengel, Robert H. Turner, Fundamentals of Thermodynamics SOLUTION MANUAL 6ed By Richard E. Sonntag, Claus Borgnakke, Gordon J. Van Wylen, chemical Engineering: Solutions for Volumes 2 and 3 by coulson 2002-12-11 In Experiments with Economic Principles, Instructor's Manual By Bergstrom Instructor Manual to An Introduction to Thermodynamics and Statistical Mechanics 2ed By Keith Stowe Instructor Manual to Introduction to Solid State Physics Eighth Edition By Charles Kittel Instructor Manual to Introductory Quantum Optics By Christopher Gerry and Peter Knight Instructor Manual to SEMICONDUCTOR DEVICES Physics and Technology Second Edition By S.M.Sze Instructor Solutions Manual to accompany Boyce Elementary Differential Equations and Boundary Value Problem 8ed by Charles W. Haines, William E. Boyce Instructor's Manual for Solving ODEs with MATLAB By L. F. Shampine, I. Gladwell, S. Thompson Instructor's Solutions for: Design of Analog CMOS Integrated Circuits by razavi Advanced Engineering Mathematics 3rd Edition by Dennis G Zill and Michael R Cullen Wireless Communication and Networks second edition William Stallings solutions manual Transport Phenomena - 2nd edition by Bird, Stewart and Lightfoot Solution Manual Signals and Systems 2nd edition simon Haykin Solutions Manual Electrical Engineering: Principles and Applications 3ed Allan R. Hambley Introduction to Chemical Engineering Thermodynamics 7th edition (solution manual) By J.M. Smith, Hendrick C Van Ness Introduction to Heat Transfer - 013391061X Solution's Manual By peyman pourmoghaddam , Vedat S. Arpaci not complete Computational Techniques for Fluid Dynamics: A Solutions Manual By Karkenahalli Srinivas, Clive A. J. Fletcher Introduction to Heat Transfer 4th Edition SOLUTION MANUAL By Frank P. Incropera, David P. DeWitt Heat Transfer-Fundamentals of Heat and Mass Transfer-Incropera & Dewitt Solution Manual power analys and design by glover 3rd ediiton Digital Communications: Fundamentals And Applications (2nd Edition)- Bernard Sklar Engineering mechanics: Dynamics - 4th Ed. by Bedford and Fowler design of machinary by norton 3rd edition Digital Signal Processing (2nd Ed.) (Mitra) Solution Manual Electronic Circuit Analysis and Design 2nd edt. by Donald A. Neamen - solution manuel Introduction to Probability by Dimitri P. Bertsekas - solution isbn: 1-886529-40-X INTRODUCTION to PROBABILITY Charles M. Grinstead and J. Laurie Snell computer networking a top down approach 3rd edition solution manual by James F.Kurose, Keith W. Ross Solution manual of sedra microelectronic circuits 5th ed. By Adel S. Sedra, Kenneth C. Smith Elementary Differential Equations And Boundary Value Problems, 7Th Ed - Boyce And Diprima Student Solutions Manual, Charles W Haines Ode Architect Companion Fundamentals of Electronic Circuit Design by David and Donal Comer Introduction to Quantum Mechanics (Second Edition) - Solutions Manual By David J. Griffiths Introductory Econometrics A Modern Approach 2ed Jeffrey Wooldridge Modern Digital and Analog Communications Systems - B P Lathi Solutions Manual 3rd edition Solutions Manual to accompany Investment Analysis & Portfolio Management, 7th edition by Reilly and Brown Selected Answers-Basic Engineering Circuit Analysis-7th Ed. by J. David Irwin Shigley's Mechanical Engineering Design 8th Ed - Solution Manual By by Richard Budynas, J. Keith Nisbett Shigley's Mechanical Engineering Design 7th Ed - Solution Manual By by Richard Budynas, J.Charles Mischke Soil Mechanics Solutions Manual (2nd Edition) By William Powrie Solution Manual for Numerical Solution of Partial Differential Equations: An Introduction 2ed By K. W. Morton, D. F. Mayers Solution Manual-Microelectronics-Digital and Analog Circuits and Systems by Millman Solutions Manual for Materials Science and Engineering An Introduction 6E By William D. Callister Solutions Manual for System Dynamics 3rd Ed By Katsuhiko Ogata Statistical Digital Signal Processing and Modeling - SOLUTIONS MANUAL By Monson H. Hayes Engineering Mechanics, statics 5th edition Solutions manual By J. L. Meriam, L. G. Kraige chapter 1-7 Probability and Statistics for Engineers and Scientists Manual HAYLER Solutions Manual Physics for Scientists and Engineers Extended Version, 5th edition, PROCESS SYSTEMS ANALYSIS AND CONTROL - DONALD R. COUGHANOWR Solution Manual Ulaby Applied Electromagnetics Butterworth Heinemann - Coulson And Richardson - Chemical Engineering Vol I (Solutions Manual V Edition) Instructor's Solutions Manual for Serway and Jewett's Physics for Scientists and Engineers Sixth Edition Volume One by Ralph V. McGrew Instructor's Solutions Manual for Serway and Jewett's Physics for Scientists and Engineers Sixth Edition Volume two by Ralph V. McGrew Digital Communications, 4th edition, 2000-08 book+solution by John Proakis Fundamentals of Wireless Communication by Tse and Viswanath Probability & Statistics for Engineers & Scientists, 8th Edition: Instructors Solution Manual ONLY by Sharon Myers , Keying Ye, Walpole Undergraduate Econometrics Solutions Manual - Hill, Judge and Griffiths CMOS Analog Circuit Design, 2ed Solutions by Phillip E. Allen, Douglas R. Holberg Solutions Manual For An Introduction To The Finite Element Method, 3rd Edition by J. N. Reddy Engineering Circuit Analysis 7ed by Hayt ISBN-10: 0071109374 ( please ask before sell or send sample) be carefullllllll Harcourt Mathematics 12 Geometry and Discrete Mathematics Solutions Manual By McGraw-Hill solutions manual for Mechanics of Materials Sixth Edition by R.C.Hibbeler by F. P. Beer, E. R. Johnston Jr., E. R. Eisenberg, & G. H. Staab Solutions Manual Vector Mechanics for EngineersSTATICS, 7th Edition by F. P. Beer, E. R. Johnston Jr., E. R. Eisenberg, & G. H. Staab Calculus Single Variable 4ed chapter 1 to 11 Hughes-Hallett, Gleason, McCallum, et al. Fourier and Laplace Transform - Antwoorden Field and Wave Electromagnetics, 2nd edition, Cheng Thomas Calculus 11th edition Multivariable Calculus: Student Solutions Manual 4th edition by James Stewart (Author) solutions manual for Introduction to Electrodynamics (Third Edition) by David J. Griffiths Engineering Mathematics, 4th Edition, Solution Manual by NEWNES APPLIED MATHEMATICS AND MODELLING FOR CHEMICAL ENGINEERS-8th Edition (solution manual) by Erwin Kreyszig Electric Machinery and Power System Fundamentals Stephen J. Chapman first edition Chemical Engineering Design, Fourth Edition: Chemical Engineering Volume 6 (Coulson & Richardson's Chemical Engineering) Student Solutions Manual and Study Guide to accompany Fundamentals of Fluid Mechanics, 5th Edition by Bruce R. Munson Donald F. Young, Theodore HARCOURT MATHEMATICS 12 Advanced Functions and Introductory Calculus - Solutions Manual by Brian / Nelson Lagrangian and Hamiltonian Mechanics Solutions to the Exercises by M. G. Calkin Linear algebra with applications 3e Otto Bretcher - Solutions Manual Soil Mechanics Concepts and Applications By William Powrie Fundamentals of Financial Management, 12th Edition (Instructors guide ONLY) by James C. Van Horne, John M Wachowic An Introduction to Numerical Analysis by Endre S.9fli, David F. Mayers Feedback Control of Dynamic Systems 4th edition Franklin - Solutions Manual Solution Manual : Nanoengineering of Structural, Functional and Smart Materials Solution Manuals +book for The Economics of Financial Markets By Roy E. Bailey Instructor's Manual Im Experiments with Economic Principles By Bergstrom Instructor Manual to A Course in Modern Mathematical Physics By Peter Szekeres Fundamentals of Chemical Reaction Engineering - Solutions Manual By Mark E. E. Davis, Robert J. J. Davis Solutions Manual for Equilibrium and Non-Equilibrium Statistical Thermodynamics By Michel Le Bellac Introduction To Electric Circuits 6th Ed [Solutions Manual] By R. C. Dorf and J. A. Svoboda Chemical Engineering Solutions manual for Volumes( 2 and 3) 3 edition Backurst J. R., Harker J.H. & Richardson J. F Fluid Mechanics with Student Resources, 5th edition 2002-12 by Frank M. White Instructor Manual to Quantum Physics Third Edition by Stephen Gasiorowicz Solutions Manual for use with Fundamentals of Corporate Finance, 4th Edition (Brealey, Myers, Marcus) by Bruce Swenson Investment Analysis and Portfolio Management- Solutions Manual, 7th Edition by Frank K. et al. Reilly Microeconomic Analysis Solution Manual - Varian 3rd edition Structural Analysis by Hibbeler 5th edition The Science and Engineering of Materials by Donald R. Askeland Frank Haddleton 4th edition solutions manual for Engineering Mechanics - Statics (11th ) by R.C.HIBBELER Differential Equations & Linear Algebra, 2nd ed., Farlow modern electronic communication 8th edition by gary miller and beasley Instructor's Manual for Solving ODEs with MATLAB By L. F. Shampine, I. Gladwell, S. Thompson Solution Manual analysis design of analog IC design Microwave And Rf Design Of Wireless Systems - Solution Manual (D M Pozar) Solution Manual Peterson and DavieÍs Computer Networks: A Systems Approach 2nd edition Solution Manual Communication Networks Fundamental concepts & key Architectures By Leon Garcia Widjaja not complete 3 4 5 6 7 8 10 Solution Manual - Electrical Machines, Drives and Power Systems 6th edition ISBN 0131969188 Logic Design and Digital Fundamentals - Moris Mano partial solutions Probability, Random Variables, and Stochastic Processes [Only Solutions Manual] by Athanasios Papoulis ,S.Unnikrishna Pillai 4th edition ntroduction to Environmental Engineering and Science (2nd Edition) (Hardcover) by Gilbert M. Masters Classical Mechanics (2nd Edition) by Herbert Goldstein Data & Computer Communications, 7th Edition [Solutions Manual only] by William Stallings Calculus Third Editon By Strauss, Bradley and Smith Fundamentals of Semiconductor Devices - Anderson SILBERSCHATZ , PETER BAER GALVIN, GREG GAGNE Electrical Machines Drives and Power Systems 6th edition by Theodore Wildi Modern Control System 9 Solution Drof Bishop Electric Machinery by A. E. Charles Kingsley, Jr.Fitzgerald 6th edition 0073660094 A First Course In Probability Solution Manual,Ross 6th Fundamentals of Electromagnetics with Engineering Applications by Stuart M. Wentworth Data Communications and Networking fourth edition by Behrouz A.Forouzan Signals and Systems, Prentice-Hall Oppenheim, Willsky, Young 2nd edition The Science and Engineering of Materials by Donald R. Askeland Frank Haddleton 4th edition FESTO DIDACTIC Process Control System Microwave Engineering 3e - David M Pozar A First Course in Abstract Algebra 7Ed - Solutions Manual by Fraleigh Addison Wesley - Distributed Systems, Concepts and Design (Exercise Solutions) - G. Coulouris, J. Dollimore and T. Kindberg allen holberg cmos analog circuit design 2nd edition homework solutions An introduction to the mathematics of financial derivatives Neftci solution manual Andrew Tanenbaum Structured Computer OrganizationSolutions Manual Bioprocess Engineering Principles - Solutions Manual (Original) by pauline m. Doran Carey, Study guide and solution manual for organic chemistry Chapra Applied Numerical Methods With Matlab For Engieers Solutions Manual Chiang & Wainwright 2005 Instructor's Manual Of Fundamental Methods Of Mathematical Economics Mc Graw Hill Computer Networks Systems Approach3ed by davie peterson solutions manual Digital image processing - Gonzalez 2Ed- Solutions Manual (209p) Doets & Eijck - The Haskell Road To Logic, Math and Programming - Solutions to Exercises Econometrics - [Instructor Solution Manual] The Econometrics of Financial Markets john y. campbell, andrew w. Lo Elements Of Information Theory - Solution Manual by thomas m. cover and joy a. Thomas Fundamentals of Actuarial Mathematicsî by D. Promislow Instructor's Manual for Sipser's Introduction to the Theory of Computation by Ching Law Instructor's Manual(Information Technology Project Management 3Rd Edition) by Kathy Schwalbe Instructor's solution manual ISBN 0534382150 A Transition to Advanced Mathematics solution manual by douglas smith 5th edition Introduction To Wireless Systems - P M Shankar - Solutions Manual Mathematics for Economists Solution Manual - Simon and Blume (ver 2) Microeconomic Analysis Third Edition by Hal R. Varian Operating System Concepts (6Th Ed)-Instructor'S Manual (A Silberschatz) Operating System Concepts, 7th Edtion, Instructor's Manual by A Silberschatz Papathomas T.V. - Solutions Manual to Accompany Millman Integrated Electronics Analog and Digital Circuits and Systems (McGraw-Hill) Pattern Recognition and Machine Learning (Solution Manual) - Bishop Solutions Manual for Statistical Inference - Casella & Berger Solutions Manual Nanoengineering of Structural Functional and Smart Materials Solutions-Mathematics for Economists Solution Manual (Blume, 1994) STRUCTURED COMPUTER ORGANIZATION 5ed ANDREW S. TANENBAUM Solution Manuals for Advanced Dynamics by Donald T. Greenwood Solution Manuals for The Economics of Financial Markets by Roy E. Bailey [Math] Haberman Applied Partial Differential Equations 4e Instructor's Manual A Quantum Approach to Condensed Matter Physics Solutions by philip l. Taylor An Introduction to Economic Dynamics 0521804787 sols Engineering Mechanics Dynamics 3rd edition solution manual Hibbeler R.C. updated fixed 09-2006 Fundamentals of Quantum Mechanics 0521829526 sols Organic Chemistry 2nd ed [Student SOLUTIONS MANUAL and Study Gde] - J. Hornback, B. Murugaverl (Thomson, 2006) WW RF Circuit Design Theory and Applications Solution Manual by reinhold ludwig pavel bretchko Solution Manual for A First Course in String Theory chapter 1 to 16 Solution manual Thermodynamics Of TurboMachinery 5th edition Solution Manual for Mathematical Models in Biology An Introduction by Elizabeth S. Allman john a. Rhodes Solution Manual for Classical Dynamics A Contemporary Approach by Jorge V. Jose, Eugene J. Saletan Solution Manual for An Introduction to Ordinary Differential Equations laser fundamentals 2nd edition by william t. Silfvast Elements of Electromagnetics, 3rd Ed., Matthew N.O. Sadiku A FIRST COURSE IN DIFFERENTIAL EQUATIONS WITH MODELING APPLICATION 7TH EDITION BY DENNIS G. ZILL Solution Manual Microelectronics Circuit Analysis Desing 3rd Edition by R. Jaeger Wiley Chemical And Engineering Thermodynamics 3Ed Solutions Manual Fracture mechanics fundamentals and applications 2nd edition anderson solution manual modern control engineering by katsuhiko ogata 4th edition Thermodynamics an engineering approach sixth edition ( SI units ) : solutions manual by Yunus A. Cengel, Michael A. Boles Fluid Mechanics - Solutions Manual by CENGEL A Guide to Physics Problems, Part 1 - Mechanics, Relativity, and Electrodynamics A Guide to Physics Problems, Part 2 - Thermodynamics, Statistical Physics, and Quantum Mechanics linear algebra Juan de Burgos Solutions Manual spanish Applied Statistics and Probability for Engineers Student 3rd.Ed Classical Electrodynamics - 2nd Ed. John David Jackson byKasper van Wijk Chemical and Engineering Thermodynamics- 3rd Edition- Solutions Manual Communication Systems Engineering Proakis J (2002) Solutions Manual Daniel Shanks - Solved And Unsolved Problems In Number Theory (2Nd Ed), 1978 Design of Analog CMOS Integrated Circuits [McGraw Hill Solutions Manual Digital Signal Processing A Computer-Based Approach 1st ed Solutions Manual mitra Elementary Mechanics & Thermodynamics [2000 by Professor Jhon W. Norbury Engineering - Materials Science, Milton Ohring Solutions Manual Fundamentals Of Fluid Mechanics 3Rd And 4Th Edition Solutions Manual Fundamentals of Thermodynamics [Sonntag-Borgnakke-Van Wylen Solutions Manual volume 1 and volume 2 Introduction to Fluid Mechanics (Fox, 5th ed) Solutions Manual Juvinall, Marshek - Fundamentals of Machine Component Design, 3rd ed - Student Solutions Manual McgrawHill - William H. Hayt, John A. Buck - Engineering Electromagnetics, 6ed Solutions Manual Physics by Resnick Halliday Krane, 5th Ed. Vol 2 Schaums Mathematical Handbook of Formulas and Tables Solution To Two-Dimensional Incompressible Navier-Stokes Equations- Maciej Matyka Prentice Hall - Solutions Manual; Communication Systems Engineering University Physics - Sears and Zemansky's 11th Ed Solutions Manual Wankat & Oreovicz - Teaching Engineering Applied Strength of Materials (4th Edition) SOLUTION MANUAL by Robert L. Mott University physics 11th edition solution manual by Young and Freedman Solution Automatic Control Systems 8Ed by Kuo and Golnaraghi Solution Manuals for Fundamentals of Quantum Mechanics For Solid State Electronics and Optics by C. L. Tang Solution Manuals for Applied Quantum Mechanics by A. F. J. Levi Digital Design (3rd Edition) by M. Morris Mano High-Speed Digital System Design A Handbook of Interconnect Theory and Design Practicesbby Stephen H. Hall IBM WebSphere RFID Handbook A Solution Guide by IBM Redbooks Solution Manual for An Introduction to Numerical Analysis by Endre Suli Solution Manual for Classical Mechanics by R. Douglas Gregory Engineering Mechanics Dynamics (11th Edition) by Russell C. Hibbeler Adaptive Control 2nd. Edt. by Karl.J.Astrom - solution manuel Applied Numerical Analysis 7Ed - Curtis F. Gerald, Patrick O. Wheatley - Solutions manual Digital Signal Processing by Thomas J. Cavicchi - solution manuel Discrete Time Signal Processing 2nd Ed - Solutions to Exercises Linear circuit analysis by R. A. DeCarlo and P. Lin - solution manuel principles and application of elect eng giorigo rizzoni Solid State Electronic Device by Ben Streetman Solution Manual Device Electronics for Integrated Circuits 3Edition Muller Kamins Solution Manual Electronic Physics Strabman Solution to Skill - Assessment Exercises to Accompany Control Systems Engineering 3rd edt. by Norman S. Nise Mechanics Of Materials Solution Manual (3Rd Ed , By Beer) nearly same 4th edition just numbered different materials science and engineering,callister 7th edition (password aguara) Computer Organization and Design: The Hardware/Software Interface, 3rd Edition by David A. Patterson, John L. Hennessy Complete solutions manual for Zill s First Course in Differential Equations with Modeling Applications (7th ed.) and Zill & amp; Cullen s Diferential Equations with Boundary-Value Problems (5th ed.) Introduction To Linear Algebra 3Ed - Gilbert Strang Solutions Manual Physics for Scientist & Engineers with Modern Physics - A strategic Approach chapter by Randall D. Knight 1-35 Signal Processing First-Mclellan, Schafer & Yoder Solution Manual chapter 3-12 Physics Principles with Applications Instructor's Solutions Manual (Giancoli, Volume 1and 2) 6th edition Anton Bivens Davis CALCULUS early transcendentals 7th edition Mechanical Vibrations - Singiresu Rao - Solutions Manual 3rd edition chapters missing 6 9 12 ( all links same ) 35 mb Numerical Methods For Engineers Solutions Manual by chapra 4th edition Computer Architecture Pipelined and Parallel Processor Design (Solutions Manual) by micheal j. flynn selected solutions Mathematical Models in Biology Sols by Elizabeth S. Allman, John A. Rhodes Dynamics of Mechanical Systems Solutions Manual (Horwood Engineering Science Series) by C. T. F. Ross Instructor Manual to Special Relativity by Patricia M. Schwarz and John H. Schwarz Solution Manuals for An Introduction to Economic Dynamics by Ronald Shone Quantum Field Theory (draft version) & Instructor's Manual by Mark Srednicki book + solution manual Solution manual for Operating systems Internals and Design principles 4th by william stallings Engineering Mechanics Statics 11th Edition By R.C.Hibbeler Solutions manual for Fundamentals of engineering thermodynamics moran shapiro Solutions Manual to accompany Corporate Finance By Stephen A. Ross 8th edition Advanced Digital Design with the Verilog HDL Michael D. Ciletti selected solutions Engineering Mechanics: Dynamics 2 Ed. by Riley and Sturges Kinetics of Catalytic Reactions--Solutions Manualby: M. Albert Vannice Solutions Manual for Techniques of Problem Solving by luiz fernandez Unit Operations of Chemical Engineering, 7th Edition, Solutions Manual by: Warren McCabe, Julian Smith, Peter Harriott Solution Manual (complete) Communication Networks Fundamental concepts & key Architectures Alberto Leon-Garcia Financial Accounting 6th edition by Harrison Solution Manual by harrison Solution Manual-Fundamentals of Digital Logic with VHDL Design-1st edition by S. Brown, Z. Vranesic Mechanics of Materials [solutions manual] hibbeler 7th edition instructor s manual with powerpoints to accompany pic microcontroller and embedded systems by muhammed ali mazidi rolin d mckinlay Accounting Concepts and Applications (9th Ed.) by W. Steve Solutions Manual to Accompany Organic Chemistry by Robert C. Atkins, Francis A Carey, Robert Atkins, Francis Carey solution manual differential equations 5th edition by zill Fundamentals of Electric Circuits 3rd edition by Alexander Sadiku Advanced Engineering Mathematics Dennis G Zill 2nd Solution solution manual Principles of corporate finance 7e Instructors Solutions Manual for Differential Equations with Boundary Value Problems, 2/E by john polking Solution Manual for Basic Engineering Circuit Analysis, 8th Edition by J. David Irwin, R. Mark Nelms Solution manual for Calculus A Complete Course 6th by R.A. Adams Elementary Linear Algebra with Applications 9 edition by Howard Anton, Chris Rorres student manual Elementary Linear Algebra with Applications 9 edition by Howard Anton, Chris Rorres Instructor Solutions Manual and Instructor Testbank Fundamentals of Engineering Thermodynamics: Si Version / 5th Edition by Michael J. Moran Howard N. Shapiro === Subject: Re: Solutions Manual for the 8th Edition of Electric Circuits by <29314065.1207401047383.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=RccSxgoAAADspVae8t3n18NSxC3vy-ig Gecko/2008060309 Firefox/3.0,gzip(gfe),gzip(gfe) Same here, I'd like a copy too. === Subject: Re: Diminishing concerns on factoring > For years I've had a major fear that if I had a major breakthrough > with integer factorization it might have a very bad, negative impact > on the global economy yes that is very true, your work is so important to everything, so tied into financial markets, except they use encryption not based upon factoring at all, > but now as I finish out the foundation layer of a foundation layer? i.e. victorian girly undergarments? > that research it is clear that my research is not impacting the > world. yep, not even impacting your world. even that imaginary one over there => look quick! >Yeah, the world is having problems but those are its own. yep, new toasters are so hard to come by. Ok, so what do I mean about the foundation layer? makeup ? the power you put on first ? Well the simplest way to consider what my factoring research does is ((((((*nothing*)))))) > that while mathematicians have traditionally focused on one type of **nope, Cryptologists, not mathematicians. > congruence I use two: 1. x^2 = y^2 mod S > 2. z^2 = y^2 mod T which reduces to a^2 = y^2 * C , where a^2 = x^2 - z^2 and C= (mod S - mod T) try to think about it. *trivial* foundation layer of surrogate factoring, and removes the need for > further brainstorming at this point, so there is no need for me to > discuss further. We are so glad you finally completed your foundation layer, and can move on up to overall color balance. Succinctly, the basic research phase on the primary layer is complete. primate layer ? Monkey Math !! You will never complete Monkey Math, it is what you do. World seems ok, as my worst fears haven't realized, .....but you haven't published anything yet..... > so I'm kicking > back, relaxing and moving to other problems as a MAJOR issue right now like where your marbles fell out ? > is monetization of the web, or so-called Web 2.0 as it's coming up as > a bigger and bigger problem to get revenue flowing for creative people > around the world, try tiny adds in the back of newspapers > and my current focus is on the biggest player, Google WILL hire you as lead independent researcher for Seirrasgoat Figuering, then sell you to Microsoft to make their sails go down. > so > YouTube is the focus of problem solving efforts and that will take up > most of my time indefinitely. Math research is done for this iteration. Actually it never started for you. > James Harris === Subject: Re: JSH: Diminishing concerns on factoring posting-account=3WPJYgoAAAA55VjhzK9i07RN8h8u8eEs Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > Ok, so what do I mean about the foundation layer? > I think the foundation layer goes on first, then blush or rouge, followed by perhaps a bit of glitter. Better ask your mom if you're unsure though, as there's nothing worse than a tacky cross-dressser. Good luck with that, M === Subject: Re: JSH: Diminishing concerns on factoring posting-account=mgs1FwoAAABD3j5T_RLZ06yrgt2dghDu Gecko/20050915,gzip(gfe),gzip(gfe) > For years I've had a major fear that if I had a major breakthrough > with integer factorization it might have a very bad, negative impact > on the global economy Yeah, a fear like that must be rough. I'll bet it's almost as bad as your fear of the fact that you'll never succeed. > Ok, so what do I mean about the foundation layer? Probably another set of vague gibberish, slightly resembling math notation, that you thought of while on the crapper. [vague gibberish, slightly resembling math notation, snipped] > a MAJOR issue right now > is monetization of the web, or so-called Web 2.0 as it's coming up as > a bigger and bigger problem to get revenue flowing for creative people > around the world Who could possibly be better at using the web to make money? Heck, just look at the boatloads of traffic you've managed to attract to your blogs! > that will take up most of my time indefinitely. Do you plan to find an appropriate newsgroup to discuss your new efforts? Or will you continue to post to the math groups? > Math research is done for this iteration. Don't let the two-dimensional tautological space hit you on the ass on your way out. -- My blogs get pathetically few hits. -- James Harris === Subject: Get instant help in Math ! posting-account=l-LiHQoAAADAkHXnI9TemrohPDDveH8c Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) We, at 24hourstutor.com, help you solve even the most intricate math problems with an exceptional ease. Our certified teachers are here to help you come over the fear that math has been able to create among students since so many years. Be it equations or graphs, calculus or trigonometry we have a solution for each and every type of algebra problem of all possible levels. At just $10/hour , you can get help from experienced teachers in a live, online voice enabled virtual classroom ! Our benefits include : 1. Guaranteed improvement in grades 2. Exclusive 1-on-1 session with the tutors 3. 24X7 availability of the tutors 4. Huge database of world's top ranked tutors - All our tutors have Masters degree in their respective subjects and a minimum of 10 years of teaching experience. 5. All the sessions are archived for future reference of the students So why wait get help now ! Log onto www.24hourstutor.com === Subject: Re: Diminishing Returns in Game Engineering AngleWyrm: > I've added a section so that it is quite clear why, called > Double-Implementation > http://home.comcast.net/~anglewyrm/diminishing_returns.htm I don't think your argumentation is correct. What you have called the top (repeating) implementation is indeed used to create this rare/common disbalance, to make outstanding things rare. A great example is Battle Isle, where units' skill level grew slower as they became more experienced. You can come to this conclusion if you assume experience is proportional to the time spent in action and that achieving each next skill level requires more experience that the previous one. If you're twice more experienced than I, you're only one step more skilled (not twice as skilled). This is kind of a logarithm, Skill = log(Experience). And know what? This has been supported by the best judge -- the reality: WWII stats show that newbie soldiers who survived their very first battle had 5-10 times higher chence to survive in the future than complete newbies. So, the growth of Skill is fast in the beginning and slows down considerably at higher levels. Logarithm behaves just like this! Now, back to the number-of-stats problem. You say increasing this number does just the same. But using your assumtion about multiplication of stats (which I don't find correct...) I have come to an opposite conclusion. Let's assume the stats grow linearly (without loss of generality, as long as they're sufficiently correlated and therefore change at approximately the same rate, as you assume): x = 0.1, x^3 = 0.001 x = 0.2, x^3 = 0.008 (+0.007) x = 0.3, x^3 = 0.027 (+0.019) ... x = 1, x^3 = 1; x = 2, x^3 = 8; (+7) x = 3, x^3 = 27; (+19) As you see, the derivative only increases, meaning that the more outstanding characteristics will be even more likely than modest ones, which is actually disgusting to nature. Is it possible that your stats improve quicker and quicker with time, without a limit? This prevents the model from converging and makes it unstable: initially a little bit more skilled units will quickly develop so emormously an advantage over those that initially were just a tiny little bit weakier, that less and less units will make sence (0.0001 against 10000 is nothing, right). And will be a self-accelerating process, so the game will be reduced to race for experience, and the winner will be known quite early... As for scaling that you say doesn't effect things, you're again not right. Number one is the turning point for the X^N function. Plot the same graphs for values from 0 to 50 and see the difference ;) ... If we stick to the scale [0,1] -- things won't be any better: Near pont point one the skill gaining speed makes a leap and significally increases. Why? In reality I don't see no reason for it, so such a model would be at least unrealisic, and at most -- not playable. Anton === Subject: Wendy Bouchoux's answers to problems from Linear Algebra 18.700-MITOCW Attached are the answers to some of the problems from MITOCW Linear Algebra 18.700. begin 666 Linear Algebra 18.700-MITOCW-Wendy Bouchoux-Answer to Problem 1 in Problem Set 3.rtf === Subject: Re: Understanding equations with multiple summations t<485A4F2E.2761D5D4@tesco.net> tposting-account=_Ec08woAAABwFaKqUMUpHCWcwTiZjSmh tGecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) Right now, Im less worried about physical implementation and more just trying to understand the equation. If I can get the theory, I can figure out the code. CHris On Jun 19, 8:21æam, Frederick Williams I suggest you ask in comp.lang.. -- > He is not here; but far away > æ The noise of life begins again > æ And ghastly thro' the drizzling rain > On the bald street breaks the blank day. === Subject: Re: co.95ncidental probability nothing could happen 100% coincidentally: there is always the possibility that, at least in 3D space, an event would happen from a cause, whether that cause be material or not. that is even if you wouldn't have any data. the notion that provided with enough data nothing in 3D space could ever happen coincidentally again is only theoretically true, or a theoretical possibility to be precise. your guess is as good as mine on this, but mine is that if space in infinitely macro, it would stand to reason that it would be infinitely micro as well, which would prevent us from ever having enough data to rule out coincidence. so the concept of coincidence doesn't need to be redefined in the subatomic, it current meaning just needs to be elaborated. it doesn't require an introducing an uncertainty principle anyway. particulartly if the matter described by it doesn't have the slightest bearing on coincidence, but on a mere physical abstraction of metaphysics entirely. === Subject: Re: co.95ncidental probability not very revolutionary perhaps, but quite elementary, if not completely fundamental or basic. I'd say the people would have a right to know. === Subject: Re: co.95ncidental probability resuming, a total probability of 1 would allways include a complement of design, the size of the design complement depending on the appearance of evidence of unlikely outcomes. the unlikelier the evidence, the greater the possibility of design. going towards a coincidental probability of zero, probability of design would go to 1. and such is the calculus of paranoid === Subject: Re: co.95ncidental probability which, by perpetual approximation, justifies Lim design + coincidence = 1 c->0 nothing mentally abject about it === Subject: Re: co.95ncidental probability correction Lim design = 1 c->0 design + coincidence = 1 in the whole domain === Subject: Re: co.95ncidental probability but of course on in the presence of evidence === Subject: Re: co.95ncidental probability ly === Subject: Re: Diminishing Returns in Game Engineering > I've added a section so that it is quite clear why, called > http://home.comcast.net/~anglewyrm/diminishing_returns.htm > This is kind of a logarithm, Skill = log(Experience). And what is a logarithm? log_x(x^y) returns y, the number of dimensions. Taking the log of something implies that it has multiple dimensions/variables. > Now, back to the number-of-stats problem. Which is another way of saying number of dimensions. > Let's assume the stats grow linearly (without loss of > generality, as long as they're sufficiently correlated and It is important that they NOT be correlated. First remove all correlation. If one stat grows at the same time that another stat grows, then they can be reworked so that there is only one growing stat, with two derived stats. > therefore change at approximately the same rate, as you assume The assumption I am making is that the center 'diagonal' produces the highest number. It could also be called the optimum price/performance ratio. > As you see, the derivative only increases, meaning that the > more outstanding characteristics will be even more likely than > modest ones ... And will be a self-accelerating process, so > the game will be reduced to race for experience, and the > winner will be known quite early... Which happens in a lot of games. Most of the 4X space games are an early race, the the entire game is decided within the opening moves. Some of them are decided even before the opening moves, by the random layout of the terrain. === Subject: Re: question about ordering fields Kevin Buzzard in litteris scripsit: > Finally, the question! Say F is a field and x is in F. > If F admits an archimedean ordering such that x becomes positive, > and F also admits a non-archimedean ordering such that x becomes negative, > then does F admit an archimedean ordering such that x becomes negative? I believe I have a counterexample. Start with Q(x) (rational functions in one indeterminate x with rational coefficients) and adjoin a square root of each polynomial x(x+n) for each positive integer n. Let F be the resulting field (just to clarify: this means, obviously, that F adjoins the square root of every polynomial of the form (x+m)(x+n) for m and n nonnegative integers). The existence of a square root guarantees that, for any ordering of F, the element x(x+n) is positive, so either x>0 either x<-n. So x must be either positive or less than any integer: so there is no archimedean ordering of F making x negative. However, taking any positive transcendental real number value for x embeds F in the reals making x positive. But we can also embed it in, say, the field of Laurent series in 1/x, which can be ordered by making x negative and less than any integer. -- David A. Madore ( http://www.madore.org/~david/ ) === Subject: On the monomial characters of the symmetric group posting-account=l8GO0wkAAAAHTcbc19NyjPKShtYYGoBR Hi I'm interested to know which irreducible characters of the symmetric group are monomial. More precisely, I want to know for which partitions $pi$, the corresponding character $chi_{pi}$ is monomial. Unfortunately, I couldn't find the answer in literature. I would be very thankful if you help me to finde a source for this problem. === Subject: Re: characters for subgroups posting-account=l8GO0wkAAAAHTcbc19NyjPKShtYYGoBR > What is the relation between characters of a group and its subgroup? e.g. what is the relation between characters of E(8) [group] and E(7) > and SU(2) [Sub-groups]? This is an important and hard problem of the representation theory, known as Branching rule. Many cases are solved for example for many classes of classical groups. === Subject: Re: Diminishing Returns in Game Engineering And what is a logarithm? log_x(x^y) returns y, the number of dimensions. Taking the log of something implies that it has multiple dimensions/variables. In my example experience is calculated as: Exp = Ta*K, K=const. -- just a value directly proportinal to the time the unit has spent fighting. It is neither multi-dimensional nor consists of several variables... The assumption I am making is that the center 'diagonal' produces the highest number. It could also be called the optimum price/performance ratio. It's a fact, not an assumption. What you assume is that the various stats are correlated (hence every point is close to the diagonal) and that the overall fittness function is a product thereof (so that being on the diagonal means being the best). I don't think either of these assumptions applies to a considerable number of games... Which happens in a lot of games. Most of the 4X space games are an early race, the the entire game is decided within the opening moves. Some of them are decided even before the opening moves, by the random layout of the terrain. said a great number of stats did the opposite thing, didn't you? The opposite situation would be if no early gained advantage could have any effect, which'd be more absurd, right? So the question is, how to find the point between the two poles: self-accelerating provess and self-deceletating process, negative or positive feedback... Common sense tells that every advantge (gained during the opening moves or in the middle of the game) should of course have an effect, but also it shuld not be too strong. What middle point is chosen is decided by how the game models the Friction Forces (as Klauzevitz called them) Another assumtion of yours that I don't like is the existence of a single fittness fucntion that summarizes a untit's effectiveness. Think of Stone-Scissors-Paper. A mojority of strategy games implement this principle, better or worse, and your 'diagonal' strategy doesn't work because the stats are negatively correlated (good against infantry but can't resisst against airctaft). So, such a unit cannot be characterized by a single statistic. This happens if pieces of magic armor offer multipliers, or a character can buff their weapon with several multipliers. True ONLY if the multipliers are correlated, but why shoul they? Also, as the number of stats increases, the product hangs around the bottom until the very end. The situation devolves into a set of boolean maxed/not-maxed prerequisites, making the individual stat ranges rather pointless. Not only this, but also the overall value of X^n starts to behave as a boolean. But that'd be so only in case of highly correlated variables, which'd mean they might be reaplaced by only one variable! In a good model the variables are not correlated and you graphs loose all sense. And replacing naturally analogous values by booleans is bad irregardless of the model and number of stats therein, because that creates lack if information. I may wiels the swor skightly better than you, and some other guy, a bit better than me. With a boolean this difference cannot be accounted for. Anton P.S.: Deciding about the number of stats before designing a game model is like choosing the number of screws before desig- ning a car. === Subject: Re: Diminishing Returns in Game Engineering > Skill = log(Experience) > It is neither multi-dimensional nor consists of several variables... The term 'log' in the above example is a bit ambiguous. Are we taking the log with a base of e, 2, 10, or some other value? Log_10 (1000) is 3. This means 10*10*10 = 1000. This is just the degenerate case of X*Y*Z. A square is still a rectangle, and a cube is still a box. If we assign a base (lets just use b and fill in the value later): Skill = log_b (Experience) means b^Skill = Experience. Or b*b*b*...(Skill times) = Experience. And perhaps this is the correlation you speak of, where the same dimension is being used multiple times. Instead of using b*b*b -- a cube, wouldn't it be more interesting to use b*c*d -- a box? > Another assumtion of yours that I don't like is the existence > of a single fitness function that summarizes a unit's > effectiveness. Such as Character Level, Armor class, Damage output, etc. Perhaps you'll invent a replacement for these. > In a good model the variables are not correlated and you > graphs loose all sense. Here is where I suggest a small change: and your graphs lose all _common_ sense. True: Common sense does not work for this problem space. > And replacing naturally analogous values by booleans is bad Unless you multiply many of them together. Because when multiplying many of them together, the result is that max dwarfs everything else. In other words, the game winds up having only one good armor/gun/whatever. > P.S.: Deciding about the number of stats before designing a game Putting the cart before the horse are we? Deciding about the number of stats _is_ designing the game, and in my opinion it is a step up from pound-and-observe. === Subject: Re: Diminishing Returns in Game Engineering AngleWyrm, The term 'log' in the above example is a bit ambiguous. Are we taking the log with a base of e, 2, 10, or some other value? Log_10 (1000) is 3. This means 10*10*10 = 1000. This is just the degenerate case of X*Y*Z. A square is still a rectangle, and a cube is still a box. The base doesn't matter much. And calling somethng just a degenerate case is a bit illogical because the particlar inherits the properties of the general, but not vice versa! A rectangle is not a square and a box is not a cube! You can't project the properties of a particular case onto the general case (at least, without applying induction correctly). And perhaps this is the correlation you speak of, where the same dimension is being used multiple times. Yes, it's absolute, 100% correlation. wouldn't it be more interesting to use b*c*d -- a box? Now that the variables are different, to assume their high correlation becomes ungrounded. Such as Character Level, Armor class, Damage output, etc. Perhaps you'll invent a replacement for these. your statements won't work with more than one because you keep thinking of the variables as highly correlated (hope you know what it means) and different functions would allow for the Scissors-Paper-Stone principle which, by definition, destroys correlation by incorporating unit's specifications, professions. Not only in strategies but also in RPGs: like a mage working on the improvement of magick skills or a warrior that has to train strenglth, stamina, weapon-wieldiing... I said: In a good model the variables are not correlated and your graphs loose all sense. And you commented: Here is where I suggest a small change: and your graphs lose all _common_ sense. True: Common sense does not work for this problem space. Why did't you react to my real criticizm instead of turning it to something totally different and commenting on the modified version, which has nothing in common with I wanted to say? Unless you multiply many of them together. Because when multiplying many of them together, the result is that max dwarfs everything else. In other words, the game winds up having only one good armor/gun/whatever. I have said why it is always bad to use booleans for analogous values. Also I have said why your logic is incorrect. Why do you repeat it here? This way you won't convince anybody, neither will you let anybody to convince you, even if you're wrong. I can only repeat that what you propose leads to nothing else than making some information not accounted for in the model. For example: S = X^4; x1 = 0.0008 x2 = 0.0009 x1/x2 = 8/9 ~ 1 S1/S2 = (8/9)^4 = 0.624 That's the difference (between S1 and S2) that you don't wonna take into consideration. Using booleans only creates noise. In other words, the game winds up having only one good armor/gun/whatever. Yes, but also a lot of bad armors/guns, which can also be compared. And if you replace them with booleans, you'll get only random noise ditributed between only two values. Putting the cart before the horse are we? Deciding about the number of stats _is_ designing the game, and in my opinion it is a step up from pound-and-observe. I think you're puttin it in front of the horse. The number of stats is only a corollary of design. Designing a game is creating a model of the game's inner physics such that it meets the designer's needs (in terms of the external, observable behavior, not the number of stats...). No matter how many stats are used. One may create a model with only two stats, and another designer -- with ten ones. And both the models will be great as long as they'll be what their authors wanted them to be. Anton === Subject: Re: Diminishing Returns in Game Engineering > This means 10*10*10 = 1000. This is just > This is just the degenerate case of X*Y*Z. > A square is still a rectangle, and a cube is still a box. > A rectangle is not a square and a box is not a cube! These two sentences do not match; the reversal was not implied. x*x*x is the degenerate case of x*y*z, where y and z happen to equal x. > The base doesn't matter much. It shows explicitly how a multiplication is taking place. Log_b( b^n ) = n. So n is the count of times b is being multiplied with itself. > It is neither multi-dimensional nor consists of several variables... It is correct to say only one variable is being used. But I cannot fully agree that it is not multi-dimensional. I argue that b^n is n-dimensional with 100% correlation between dimensions > Instead of using b*b*b -- a cube, > wouldn't it be more interesting to use b*c*d -- a box? > Now that the variables are different, to assume their high > correlation becomes ungrounded. Perhaps we should use random numbers, to make the point clear. Let x and y be random linear variables in the range 0..1. So by definition, there is a 0% correlation between x and y. Plotting points (x,y) fills a plane with random noise. Now let z = x * y, and plot in three dimensions. The unit cube is not filled with random noise, but instead displays this graph: http://home.comcast.net/~anglewyrm/thread/f(xy).png The curved central diagonal line is the degenerate (and optimal) b*b*b... case within the surface of all b*c*d... possibilities. > Such as Character Level, Armor class, Damage output, etc. > Perhaps you'll invent a replacement for these. Sorry, and this is really good information for me, thank you. I will have to fix the page on this matter, so that it's clear about what kind of situations the subject applies to. > [both] models will be great as long as they'll be what their authors > wanted them to be. Wish that were true more often, which is why I'm working on this page. === Subject: Re: Diminishing Returns in Game Engineering <862dnVC3MJN6BcjVnZ2dnUVZ_oLinZ2d@comcast.com> posting-account=AyHWCwoAAABogJDv7KM_w1svn59mI0Nd Gecko/2008061015 Firefox/3.0,gzip(gfe),gzip(gfe) I still cannot explain myself why does anybody lose time trying to explain to this AngleWyrm things he clearly does not want (and anyway cannot) understand. Don't you all see he still does not understand what a logarithmic trend is? Probably he used wikipedia to understand the definition of Log and came here to explain us and propose the usage of that box model. === Subject: Re: Diminishing Returns in Game Engineering <862dnVC3MJN6BcjVnZ2dnUVZ_oLinZ2d@comcast.com> posting-account=AyHWCwoAAABogJDv7KM_w1svn59mI0Nd Gecko/2008061015 Firefox/3.0,gzip(gfe),gzip(gfe) Ah, i forgot: Sorry AngleWyrm but this is the hard trouth. studied and developed enough. Maybe if you develop a simple textual game (cards game or similar) in two versions: the wrong and the right way in your opinion you will not understand yourself good enough to explain others. === Subject: Re: Diminishing Returns in Game Engineering <862dnVC3MJN6BcjVnZ2dnUVZ_oLinZ2d@comcast.com> posting-account=db483goAAAC0wMFPxnsgaSY1yXkCYGTR 1.0.3705; .NET CLR 1.1.4322; Media Center PC 4.0; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 3.0.04506.590),gzip(gfe),gzip(gfe) æAnd what is a logarithm? log x(x^y) returns y, the ænumber æof > æ dimensions. Taking the log of something implies æthat æit æhas > æ multiple dimensions/variables. æ In my example experience is calculated as: > æ Exp = Ta*K, K=const. > æ -- just a value directly proportinal to the time the unit æhas > æ spent fighting. It is neither multi-dimensional ænor æconsists > æ of several variables... æThe assumption I am æmaking æis æthat æthe æcenter æ'diagonal' > æ produces the highest number. æIt æcould æalso æbe æcalled æthe > æ optimum price/performance ratio. æ It's a fact, not an assumption. What you assume æis æthat æthe > æ various stats are correlated (hence every point æis æclose æto > æ the diagonal) and that the æoverall æfittness æfunction æis æa > æ product thereof (so that being on the diagonal means being the > æ best). I don't think either of these assumptions applies to æa > æ considerable number of games... I'd save my breath, facts are not going to sway him. The ending point for me was when he 'creatively edited' one of my replies, snipping out his mistake and most of my response to it, until he was left with an appropriately contextless question that he could use to wave vaguely at his web page and insist the answer was in there. (Actually, the first time I saw this method used was by Wally in a Dilbert cartoon. Next step, IIRC, is to claim that the person misunderstood their own question.) My personal guess is that either he either doesn't have the patience, or is just not skillful enough at this style of game, and the whole purpose of the page is to shift the blame to someone else. The premise is flawed, the math questionable, the logic laughable, the page is downright embarassing for its creator. But like most crackpots, I'm sure he'll insist that it is correct and shift the blame to us for not 'getting it'. ----- Geoff === Subject: Re: co.95ncidental probability more specifically, if we don't care to know the cause, i.e. to calculate it. so coincidence is basicly a product of incapacity. that is, of course, disregarding the endless philosphical efforts attempting to define the concept of cause and effect in 18th century europe, most of which is dealing with god rather than with the universe anyway. and god doesn't seem to be the cause of anything. so let's not confuse design with the total set of all causes in the universe, i.e. the universe itself previous to an event, in other words let's not confuse design with the universe itself, which may be frought with all conceivable colours of purpose for all I care, but we simply do not know, nor do we have any way of knowing. === Subject: Re: co.95ncidental probability it seems to be a rather ideal way of constructing would-be information. information that is largely metaphysical, i.e. that doesn't bear the slightest trace of anything contingental, that is, real. === Subject: Re: co.95ncidental probability it doesn't seem to be very useful to test the things you already know, and still this is what current probability theory seems to be all about. === Subject: Re: co.95ncidental probability so it's not metaphysical design, but design that can at least theoretically be actually established or measured and that is therefore quite physical, and that therefore should belong in probability theory. anyway, fundamental concepts in probability theory seem to be events, data, evidence, coincidence and design, and perhaps chaos. how these relate exactly I don't know, or I would have to brainstorm something rightaway, but a question that would deserve an answer would be if data and evidence aren't in fact identical concepts, n.b. in probability theory. data the way I put it is enough knowledge about the cause of an event to stop considering (calling) it coincidental. whereas evidence would be events itself, specifically those supporting a suspicion of design. but outside of this framework evidence is of course data as well, and perhaps it would be impossible to gather data in the sense of the framework in another way than by producing evidence in the sense of the framework, i.e. by doing tests. only physical tests seem to be somewhat more useful than the largely confirmatory tests that probability theory does. it's like testing the metaphysical and as we know you cannot test the metaphysical and expect to get the unexpected because, it entirely comprises, consists of and equals the set of all things tautologous. the metaphysical is the complementary set of the contingent. as to coincidence and design, once we have enough knowledge about the cause of evidence to stop calling it coincidental, it would of course not necessarily be design. it would either be due to the normal chaotic behaviour of 3D space, or it would be in such a way singular or regular that it would be indicative of a more regular cause or origin, and the only thing regular in all of space is life. so what event or evidence is coincidental without knowledge or data, with such data is either chaoticly defined or not. if it's not, it would be alive, and the cause would be design. === Subject: Re: co.95ncidental probability the design I'm talking about is beyond god entirely. === Subject: Re: co.95ncidental probability eh, doesn't stoop === Subject: Re: co.95ncidental probability its current meaning just needs to be elaborated, i.e. coincidence is not a logical fundamental, but a name we give to the cause of events if we do not know it. === Subject: Re: Instant solutions to all Math Problems [...] Be it equations or graphs we have a > solution for each and every type of math problem of all levels. Really? -- He is not here; but far away The noise of life begins again And ghastly thro' the drizzling rain On the bald street breaks the blank day. === Subject: JSH: Probabilistic quadratic residue solving posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) An off-shoot of my surrogate factoring research is a probabilistic method to solve quadratic residues, as given k^2 = q mod p when p is an odd prime, and q is a quadratic residue modulo p, you find k. The technique requires introduction of a few additional variables starting with T, where T = 2q + np where n is an odd natural number so you have T = 2q mod p, but T - 2q is also forced to have p as a factor. Next you find z, where with integer factors f_1 and f_2 where f_1*f_2 = T: z = (f_1 + f_2)/2 and now finally you try for an answer for k, with k = 3^{-1}(2z) mod p. The method is probabilistic because if I've got the analysis right you have a 50% probability of getting the right k, for each z that you try. Checking is done by looking at k^2 mod p, to see if you get q. Example: Let q=2, p=17 so T = 2(2) mod 17 = 4 mod 17. Here T=21 does not work, but T = 55 = 5(11), so z = 8 and the answer then from 3k = 2(8) mod 17, is k = 11 mod 17. Is there any use for such a technique? James Harris === Subject: Re: Probabilistic quadratic residue solving JSH > An off-shoot of my surrogate factoring research is a probabilistic > method to solve quadratic residues, as given k^2 = q mod p when p is an odd prime, and q is a quadratic residue modulo p, you > find k. The technique requires introduction of a few additional variables LOL === Subject: Re: JSH: Probabilistic quadratic residue solving posting-account=a6woBRAAAADpNFZJBA7ZBx35zXaKmaP4 Gecko/2008061017 Firefox/3.0,gzip(gfe),gzip(gfe) > The method is probabilistic because if I've got the analysis right you > have a 50% probability of getting the right k, for each z that you > try. æChecking is done by looking at k^2 mod p, to see if you get q. Example: Let q=2, p=17 so T = 2(2) mod 17 = 4 mod 17. Here T=21 does not work, but T = 55 = 5(11), so z = 8 and the answer > then from 3k = 2(8) mod 17, is k = 11 mod 17. Is there any use for such a technique? James Harris For primes p = 8n + 1, there is a probabilistic aspect of the best methods for finding square roots mod p. For primes with other residues mod 8, deterministic methods are known. However the nondeterminism is limited to finding one quadratic residue mod p. If only one square root mod p is needed, then perhaps the analysis comes to the same thing, but it is well known that half of all the nonzero residues mod p are quadratic nonresidues, and confirming one by Legendre symbol (efficiently found by the generalization, the Jacobi symbol) chosen at random clearly gives a 50% chance of success. Shanks-Tonelli algorithm would be nice starting points if you wish know more. === Subject: Re: JSH: Probabilistic quadratic residue solving posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) > The method is probabilistic because if I've got the analysis right you > have a 50% probability of getting the right k, for each z that you > try. æChecking is done by looking at k^2 mod p, to see if you get q. Example: Let q=2, p=17 so T = 2(2) mod 17 = 4 mod 17. Here T=21 does not work, but T = 55 = 5(11), so z = 8 and the answer > then from 3k = 2(8) mod 17, is k = 11 mod 17. Is there any use for such a technique? James Harris For primes p = 8n + 1, there is a probabilistic aspect > of the best methods for finding square roots mod p. > For primes with other residues mod 8, deterministic > methods are known. Well this technique would be probabilistic for p odd prime. So it's more general then, correct? > However the nondeterminism is limited to finding one > quadratic residue mod p. æIf only one square root mod > p is needed, then perhaps the analysis comes to the > same thing, but it is well known that half of all the > nonzero residues mod p are quadratic nonresidues, and > confirming one by Legendre symbol (efficiently found > by the generalization, the Jacobi symbol) chosen at > random clearly gives a 50% chance of success. Um, yeah. Already read it. > Shanks-Tonelli algorithm would be nice starting > points if you wish know more. I think it interesting to me that what I call surrogate factoring leads to this way to solve for a quadratic residue by factoring. Note my idea involves factoring an odd T chosen such that T = 2q mod p, where q is the quadratic residue modulo p, and it should have a 50% chance of success with each factorization of T, if it's right. The math is easy but repeatedly I've thought one thing and been forced to back down later when experiments demonstrate that I'm wrong. Still I find myself drawn in a bit on this subject so I'm reading material about it on the web and pondering the problem. JSH === Subject: Re: JSH: Probabilistic quadratic residue solving > The method is probabilistic because if I've got the analysis right you > have a 50% probability of getting the right k, for each z that you > try. æChecking is done by looking at k^2 mod p, to see if you get q. > Example: Let q=2, p=17 so T = 2(2) mod 17 = 4 mod 17. > Here T=21 does not work, but T = 55 = 5(11), so z = 8 and the answer > then from > 3k = 2(8) mod 17, is k = 11 mod 17. > Is there any use for such a technique? > James Harris > For primes p = 8n + 1, there is a probabilistic aspect > of the best methods for finding square roots mod p. > For primes with other residues mod 8, deterministic > methods are known. Well this technique would be probabilistic for p odd prime. So it's more general then, correct? If you want a more general probabilistic algorithm, then yes. If you want a deterministic algorithm then no. For myself I use the deterministic algorithms (p mod 8 = 3, 5, 7) where possible and only use the probabilistic algorithm (p mod 8 = 1) where I have to. Crandall and Pomerance, algorithm 2.3.8 refers. YMMV. > However the nondeterminism is limited to finding one > quadratic residue mod p. æIf only one square root mod > p is needed, then perhaps the analysis comes to the > same thing, but it is well known that half of all the > nonzero residues mod p are quadratic nonresidues, and > confirming one by Legendre symbol (efficiently found > by the generalization, the Jacobi symbol) chosen at > random clearly gives a 50% chance of success. Um, yeah. >Already read it. > Shanks-Tonelli algorithm would be nice starting > points if you wish know more. I think it interesting to me that what I call surrogate factoring >leads to this way to solve for a quadratic residue by factoring. Note >my idea involves factoring an odd T chosen such that T = 2q mod p, >where q is the quadratic residue modulo p, and it should have a 50% >chance of success with each factorization of T, if it's right. The math is easy but repeatedly I've thought one thing and been forced >to back down later when experiments demonstrate that I'm wrong. You should consider this observation well James. rossum >Still I find myself drawn in a bit on this subject so I'm reading >material about it on the web and pondering the problem. >___JSH > === Subject: Re: JSH: Probabilistic quadratic residue solving Mail-To-News-Contact: abuse@dizum.com Our chief resident crank pondered > An off-shoot of my surrogate factoring research is > a probabilistic method to solve quadratic residues, > as given (snip some of his usual crap) > Is there any use for such a technique? To give you something to further clutter up these NGs with unmitigated fecal analysis? But we still luvya anyways!!! Xs & Os Il mittente di questo messaggio|The sender address of this non corrisponde ad un utente |message is not related to a real reale ma all'indirizzo fittizio|person but to a fake address of an di un sistema anonimizzatore |anonymous system Per maggiori informazioni |For more info https://www.mixmaster.it === > For years I've had a major fear that if I had a major breakthrough > with integer factorization it might have a very bad, negative impact > on the global economy but now as I finish out the foundation layer of > that research it is clear that my research is not impacting the > world. æYeah, the world is having problems but those are its own. > Becuase you HAVE no result, that's why! You were challenged NUMEROUS times to dramatically demonstrate your result by factoring a very large integer and you *never* did so. Because of that, why should anyone be bothered to believe your claim? Once again, I would like you to tell me what the factorization is of any of those RSA numbers. Any one that hasn't been factored yet and is too big to be factored with conventional methods on today's computers. Do that and you'll have everyone a believer. Otherwise, your claim is going to get nowhere . You say you have the algorithm. Now why not go and implement it in a program, and run it over an RSA number? === Subject: Re: JSH: Diminishing concerns on factoring > For years I've had a major fear that if I had a major breakthrough > with integer factorization it might have a very bad, negative impact > on the global economy but now as I finish out the foundation layer of > that research it is clear that my research is not impacting the > world. æYeah, the world is having problems but those are its own. > That was then and I still don't see how I'm impacting the world yet, but my concerns have returned as I've solved the problem of finding the odd prime p. > Ok, so what do I mean about the foundation layer? Well the simplest way to consider what my factoring research does is > that while mathematicians have traditionally focused on one type of > congruence I use two: 1. æx^2 = y^2 mod S > 2. æz^2 = y^2 mod T where T is your target composite and S is what I call the surrogate, > and the concept I call surrogate factoring. A little while back rather than consider S as a whole I started > focusing on its prime factors, which is why I have a lot of equations > with x^2 = y^2 mod p as p is an odd prime factor of S. A major problem as T increased in size though was, how do you pick p? The larger the odd prime you used the better, but with larger and larger primes the odds of picking one that would work dropped. So I concentrated on using small primes with the Chinese Remainder Theorem, but there was the issue of false positives (which may or may not be resolvable as I didn't continue experimental research in that area). Now though I think there may be a simpler answer. > That was a HUGE breakthrough to move from concentrating on S itself, > where I had control variables I called n, alpha and k, and would > puzzle over things like what was the way to pick k, to figuring out > how to get p. And with the fundamental equations now known that completes the > foundation layer of surrogate factoring, and removes the need for > further brainstorming at this point, so there is no need for me to > discuss further. > I'm evaluating the situation this weekend as I ponder what to do next. It's a thorny problem with a lot of complexity introduced by the continuing refusal of the mathematical community to acknowledge this research. The simplest answer, if possible, may be to develop a working program and turn it over the United States Government. I would suggest that mathematicians around the world re-think their positions on this issue. My major concern is that I will end up giving up on my dream of restoring real mathematical research in number theory and leaping humanity forward mathematically, and instead find myself at the heart of military endeavors as my problem solving skills are turned instead to activities that are invisible to the world. If that happens none of your countries will have any chance of competing with the US within your lifetimes or the foreseeable future, and much of what I do will be invisible, but the impact will be unstoppable. James Harris === Subject: Re: JSH: Diminishing concerns on factoring For years I've had a major fear that if I had a major breakthrough > with integer factorization it might have a very bad, negative impact > on the global economy but now as I finish out the foundation layer of > that research it is clear that my research is not impacting the > world. æYeah, the world is having problems but those are its own. That was then and I still don't see how I'm impacting the world yet, > but my concerns have returned as I've solved the problem of finding > the odd prime p. > Without a real demonstration the only assumption that can reasonably be made is your claim Has No Substance To It!!!!!! === Subject: Re: JSH: Diminishing concerns on factoring posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) > For years I've had a major fear that if I had a major breakthrough > with integer factorization it might have a very bad, negative impact > on the global economy but now as I finish out the foundation layer of > that research it is clear that my research is not impacting the > world. æYeah, the world is having problems but those are its own. That was then and I still don't see how I'm impacting the world yet, > but my concerns have returned as I've solved the problem of finding > the odd prime p. Without a real demonstration the only assumption that can reasonably > be made is your claim Has No Substance To It!!!!!! Why? What would a demonstration prove? JSH === Subject: Re: JSH: Diminishing concerns on factoring SV1),gzip(gfe),gzip(gfe) For years I've had a major fear that if I had a major breakthrough > with integer factorization it might have a very bad, negative impact > on the global economy but now as I finish out the foundation layer of > that research it is clear that my research is not impacting the > world. æYeah, the world is having problems but those are its own. That was then and I still don't see how I'm impacting the world yet, > but my concerns have returned as I've solved the problem of finding > the odd prime p. Without a real demonstration the only assumption that can reasonably > be made is your claim Has No Substance To It!!!!!! Why? æWhat would a demonstration prove? JSH- Hide quoted text - - Show quoted text - It would prove that you have something worthwhile. === Subject: Re: JSH: Diminishing concerns on factoring posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) > For years I've had a major fear that if I had a major breakthrough > with integer factorization it might have a very bad, negative impact > on the global economy but now as I finish out the foundation layer of > that research it is clear that my research is not impacting the > world. æYeah, the world is having problems but those are its own. That was then and I still don't see how I'm impacting the world yet, > but my concerns have returned as I've solved the problem of finding > the odd prime p. Without a real demonstration the only assumption that can reasonably > be made is your claim Has No Substance To It!!!!!! Why? æWhat would a demonstration prove? JSH- Hide quoted text - - Show quoted text - It would prove that you have something worthwhile. So what if I don't demonstrate with a factorization of a very large number? What does that prove? JSH === Subject: Re: JSH: Diminishing concerns on factoring posting-account=aLpfCwoAAACh4BOs3HOlQBCoxUpEgyxc Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) Without a real demonstration the only assumption that can reasonably > be made is your claim Has No Substance To It!!!!!! Why? What would a demonstration prove? It would prove that you have something worthwhile. So what if I don't demonstrate with a factorization of a very large > number? What does that prove? It proves that you don't learn from your mistakes. === Subject: Re: JSH: Diminishing concerns on factoring posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) > Without a real demonstration the only assumption that can reasonably > be made is your claim Has No Substance To It!!!!!! Why? æWhat would a demonstration prove? It would prove that you have something worthwhile. So what if I don't demonstrate with a factorization of a very large > number? æWhat does that prove? It proves that you don't learn from your mistakes. Yawn. And you do? If so, why are you replying to me now? If I could demonstrate with a large factorization I would have done it. I can't. James Harris === Subject: Re: JSH: Diminishing concerns on factoring It proves that you don't learn from your mistakes. >Yawn. And you do? If so, why are you replying to me now? >If I could demonstrate with a large factorization I would have done >it. >I can't. >James Harris Your admission proves you are only a troll, a one trick phony. === Subject: Re: JSH: Diminishing concerns on factoring posting-account=JpxxPAgAAAAgwzQIYqn4j6syK-YhOmcF Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) > The simplest answer, if possible, may be to develop a working program > and turn it over the United States Government. > what a great idea! may I recommend to develop the following program: print Hello world! The impact of such program on the world will be truly unstoppable. === Subject: Re: JSH: Diminishing concerns on factoring > The simplest answer, if possible, may be to develop a working program > and turn it over the United States Government. what a great idea! may I recommend to develop the following program: print Hello world! The impact of such program on the world will be truly unstoppable. I think not. The current US Administration might well see such a program as dangerously close to expressing support for the United Nations. A program that printed Hello America! would be more likely to be accepted. rossum === === Subject: solution manual Engineering Mechanics Statics 6th edition Can anyone email me the solution manual for Engineering Mechanics Statics 6th edition by J.L Meriam and L.G. Kraige. I would really === Subject: hello i need the solutions of Foundations of Electromagnetic Theory.3rd edition.Reitz,Milford,Christy but i'm a students, so i do not have money to pay... could somebody helpme === Subject: hello i need the solutions of Foundations of Electromagnetic Theory.3rd edition.Reitz,Milford,Christy but i'm a students, so i do not have money to pay... could somebody helpme === Subject: Re: hello > i need the solutions of Foundations of Electromagnetic Theory.3rd > edition.Reitz,Milford,Christy but i'm a students, so i do not have > money to pay... could somebody helpme You can't get anything for free anymore, sorry. Electromagnetic theory is not that hard to understand though. The text that you mention is pretty good too. Just study it and you shouldn't need a solutions manual. Patrick === Subject: Probability, Random Variables and Random Signal Principles (4th Ed., Peyton Z., Jr. Peebles) Answers Hi there, i am desperately in need of the answers for exercises in Probability, Random Variables and Random Signal Principles (4th Ed., Peyton Z., Jr. Peebles). The book consists of abundances of exercises but there in no answers, (leave alone solution manual). This has been very annoying since i cant check my answers, i do not know if what im doing is correct or not. I have been googling it for one week now but i have been chasing my own tail. This is my last resort, i hope anyone === Subject: JSH: Probabilistic quadratic residue solving posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) An off-shoot of my surrogate factoring research is a probabilistic method to solve quadratic residues, as given k^2 = q mod p when p is an odd prime, and q is a quadratic residue modulo p, you find k. The technique requires introduction of a few additional variables starting with T, where T = 2q + np where n is an odd natural number so you have T = 2q mod p, but T - 2q is also forced to have p as a factor. Next you find z, where with integer factors f_1 and f_2 where f_1*f_2 = T: z = (f_1 + f_2)/2 and now finally you try for an answer for k, with k = 3^{-1}(2z) mod p. The method is probabilistic because if I've got the analysis right you have a 50% probability of getting the right k, for each z that you try. Checking is done by looking at k^2 mod p, to see if you get q. Example: Let q=2, p=17 so T = 2(2) mod 17 = 4 mod 17. Here T=21 does not work, but T = 55 = 5(11), so z = 8 and the answer then from 3k = 2(8) mod 17, is k = 11 mod 17. Is there any use for such a technique? James Harris === Subject: Re: Probabilistic quadratic residue solving JSH > An off-shoot of my surrogate factoring research is a probabilistic > method to solve quadratic residues, as given k^2 = q mod p when p is an odd prime, and q is a quadratic residue modulo p, you > find k. The technique requires introduction of a few additional variables LOL === Subject: Re: JSH: Probabilistic quadratic residue solving posting-account=a6woBRAAAADpNFZJBA7ZBx35zXaKmaP4 Gecko/2008061017 Firefox/3.0,gzip(gfe),gzip(gfe) > The method is probabilistic because if I've got the analysis right you > have a 50% probability of getting the right k, for each z that you > try. æChecking is done by looking at k^2 mod p, to see if you get q. Example: Let q=2, p=17 so T = 2(2) mod 17 = 4 mod 17. Here T=21 does not work, but T = 55 = 5(11), so z = 8 and the answer > then from 3k = 2(8) mod 17, is k = 11 mod 17. Is there any use for such a technique? James Harris For primes p = 8n + 1, there is a probabilistic aspect of the best methods for finding square roots mod p. For primes with other residues mod 8, deterministic methods are known. However the nondeterminism is limited to finding one quadratic residue mod p. If only one square root mod p is needed, then perhaps the analysis comes to the same thing, but it is well known that half of all the nonzero residues mod p are quadratic nonresidues, and confirming one by Legendre symbol (efficiently found by the generalization, the Jacobi symbol) chosen at random clearly gives a 50% chance of success. Shanks-Tonelli algorithm would be nice starting points if you wish know more. === Subject: Re: JSH: Probabilistic quadratic residue solving posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) > The method is probabilistic because if I've got the analysis right you > have a 50% probability of getting the right k, for each z that you > try. æChecking is done by looking at k^2 mod p, to see if you get q. Example: Let q=2, p=17 so T = 2(2) mod 17 = 4 mod 17. Here T=21 does not work, but T = 55 = 5(11), so z = 8 and the answer > then from 3k = 2(8) mod 17, is k = 11 mod 17. Is there any use for such a technique? James Harris For primes p = 8n + 1, there is a probabilistic aspect > of the best methods for finding square roots mod p. > For primes with other residues mod 8, deterministic > methods are known. Well this technique would be probabilistic for p odd prime. So it's more general then, correct? > However the nondeterminism is limited to finding one > quadratic residue mod p. æIf only one square root mod > p is needed, then perhaps the analysis comes to the > same thing, but it is well known that half of all the > nonzero residues mod p are quadratic nonresidues, and > confirming one by Legendre symbol (efficiently found > by the generalization, the Jacobi symbol) chosen at > random clearly gives a 50% chance of success. Um, yeah. Already read it. > Shanks-Tonelli algorithm would be nice starting > points if you wish know more. I think it interesting to me that what I call surrogate factoring leads to this way to solve for a quadratic residue by factoring. Note my idea involves factoring an odd T chosen such that T = 2q mod p, where q is the quadratic residue modulo p, and it should have a 50% chance of success with each factorization of T, if it's right. The math is easy but repeatedly I've thought one thing and been forced to back down later when experiments demonstrate that I'm wrong. Still I find myself drawn in a bit on this subject so I'm reading material about it on the web and pondering the problem. JSH === Subject: Re: JSH: Probabilistic quadratic residue solving > The method is probabilistic because if I've got the analysis right you > have a 50% probability of getting the right k, for each z that you > try. æChecking is done by looking at k^2 mod p, to see if you get q. > Example: Let q=2, p=17 so T = 2(2) mod 17 = 4 mod 17. > Here T=21 does not work, but T = 55 = 5(11), so z = 8 and the answer > then from > 3k = 2(8) mod 17, is k = 11 mod 17. > Is there any use for such a technique? > James Harris > For primes p = 8n + 1, there is a probabilistic aspect > of the best methods for finding square roots mod p. > For primes with other residues mod 8, deterministic > methods are known. Well this technique would be probabilistic for p odd prime. So it's more general then, correct? If you want a more general probabilistic algorithm, then yes. If you want a deterministic algorithm then no. For myself I use the deterministic algorithms (p mod 8 = 3, 5, 7) where possible and only use the probabilistic algorithm (p mod 8 = 1) where I have to. Crandall and Pomerance, algorithm 2.3.8 refers. YMMV. > However the nondeterminism is limited to finding one > quadratic residue mod p. æIf only one square root mod > p is needed, then perhaps the analysis comes to the > same thing, but it is well known that half of all the > nonzero residues mod p are quadratic nonresidues, and > confirming one by Legendre symbol (efficiently found > by the generalization, the Jacobi symbol) chosen at > random clearly gives a 50% chance of success. Um, yeah. >Already read it. > Shanks-Tonelli algorithm would be nice starting > points if you wish know more. I think it interesting to me that what I call surrogate factoring >leads to this way to solve for a quadratic residue by factoring. Note >my idea involves factoring an odd T chosen such that T = 2q mod p, >where q is the quadratic residue modulo p, and it should have a 50% >chance of success with each factorization of T, if it's right. The math is easy but repeatedly I've thought one thing and been forced >to back down later when experiments demonstrate that I'm wrong. You should consider this observation well James. rossum >Still I find myself drawn in a bit on this subject so I'm reading >material about it on the web and pondering the problem. >___JSH > === Subject: Re: JSH: Probabilistic quadratic residue solving Mail-To-News-Contact: abuse@dizum.com Our chief resident crank pondered > An off-shoot of my surrogate factoring research is > a probabilistic method to solve quadratic residues, > as given (snip some of his usual crap) > Is there any use for such a technique? To give you something to further clutter up these NGs with unmitigated fecal analysis? But we still luvya anyways!!! Xs & Os Il mittente di questo messaggio|The sender address of this non corrisponde ad un utente |message is not related to a real reale ma all'indirizzo fittizio|person but to a fake address of an di un sistema anonimizzatore |anonymous system Per maggiori informazioni |For more info https://www.mixmaster.it === Subject: Re: Algebra. > Indeed you're probabily right.My english is not > best,cause as mathmatitiam i only speake my mothe > tongue. > I said justify.. But this one is for you... A dice has in its faces the numbers 1 to 6,but the > e probability in one toss of a face stay turned up is > directly proporcional to number writen on it. Saying the probability is proportional to the number means that if the probability a 1 comes up is k, then the probabilty the number 2 comes up is 2k, the probability that the number 3 comes up is 3k, etc. Since the total of all probabilities must be 1, you have k+ 2k+ 3k+ 4k+ 5k+ 6k= 1 or k(1+ 2+ 3+ 4+ 5+ 6)= k(21)= 1 so k= 1/21. The probablity of rolling a 1 is 1/21, a 2 is 2/21, etc. Calculate the prob of multiple of three in one > e toss. > a multiple of 3 in one toss means either a 3 or a 6. The probability of a 3 is 3/21= 1/7 and the probability of a 6 is 6/21= 2/78. The probability of getting one or the other is 3/21+ 6/21= 1/7+ 2/7= 3/7. When we toss a dice consider number inferior to 5 > Determine the probability that the the happening > verifies itself at least three times in 10 tosses. > inferior to 5 (I would say less than 5) means 1, 2, 3, or 4. The probaility of that in one throw is 1/21+ 2/21+ 3/21+ 4/21= 10/21. The probability of that NOT happening (getting a 5 or 6) is 1- 10/21= 11/21. Happens at least 3 times in 10 tosses is the same as does NOT happen 0, 1 or 2 times and so the probablity of Happens at least 3 times in 10 tosses is 1 minus the probability happens exactly 0 or 1 or 2 times. Happens 0 times means getting a 5 or a 6 on EVERY toss: (11/21)^10 . Happens exactly once might be, for example, happens on the first throw but not the next 9: (10/21)(11/21)^9. But then we need to consider the possibility of happening only on the second throw, only on the third throw, etc. Fortunately, it is easy to see that those all have the same probability: (10/21)(11/21)^9 and there are 10 such cases: the probability of getting exactly 1 less than 5 in 10 throws is 10(10/21)(11/21)^9. The probability of getting less than 5 on the FIRST two throws and 5 or larger on the other 8 is (10/21)^2(11/21)^8. But now we have to consider the possibility of less than 5 on first and third etc. Fortunately it is again easy to see that the probability of each of those is (10/21)^2(11/21)^8 and also that the total number of such combinations is the binomial coefficient 2C10= 10!/2!(8!)= 10*9/2= 5*9= 45. The probability of getting less than 5 twice and 5 or greater the other 8 times is 45(10/21)^2(11/21)^8. The probability of getting 0 or 1 or 2 throws of less than 5 is the sum of those and the probability of getting 3 or more throws less than 5 is 1 minus that sum. SO,THIS IS ENDING UP IN JAKARTA...OK BY THAT TIME,WHEN THEY LOST THE GENERAL IN THE > E BATTLE FIELD,IT WAS AN IRONY TO BE A TEACHER...WHAT > WOULD HAVE TO BE DONE TO REFORM THE EDUCATIONAL > SYSTEM.... === Subject: Re: JSH: Diminishing concerns on factoring posting-account=fwSgtAkAAACFnX70ssKwbvm9_oCZVHrx 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > For years I've had a major fear that if I had a major breakthrough > with integer factorization it might have a very bad, negative impact > on the global economy but now as I finish out the foundation layer of > that research it is clear that my research is not impacting the > world. æYeah, the world is having problems but those are its own. > Becuase you HAVE no result, that's why! You were challenged NUMEROUS times to dramatically demonstrate your result by factoring a very large integer and you *never* did so. Because of that, why should anyone be bothered to believe your claim? Once again, I would like you to tell me what the factorization is of any of those RSA numbers. Any one that hasn't been factored yet and is too big to be factored with conventional methods on today's computers. Do that and you'll have everyone a believer. Otherwise, your claim is going to get nowhere . You say you have the algorithm. Now why not go and implement it in a program, and run it over an RSA number? === Subject: Re: JSH: Diminishing concerns on factoring posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) > For years I've had a major fear that if I had a major breakthrough > with integer factorization it might have a very bad, negative impact > on the global economy but now as I finish out the foundation layer of > that research it is clear that my research is not impacting the > world. æYeah, the world is having problems but those are its own. > That was then and I still don't see how I'm impacting the world yet, but my concerns have returned as I've solved the problem of finding the odd prime p. > Ok, so what do I mean about the foundation layer? Well the simplest way to consider what my factoring research does is > that while mathematicians have traditionally focused on one type of > congruence I use two: 1. æx^2 = y^2 mod S > 2. æz^2 = y^2 mod T where T is your target composite and S is what I call the surrogate, > and the concept I call surrogate factoring. A little while back rather than consider S as a whole I started > focusing on its prime factors, which is why I have a lot of equations > with x^2 = y^2 mod p as p is an odd prime factor of S. A major problem as T increased in size though was, how do you pick p? The larger the odd prime you used the better, but with larger and larger primes the odds of picking one that would work dropped. So I concentrated on using small primes with the Chinese Remainder Theorem, but there was the issue of false positives (which may or may not be resolvable as I didn't continue experimental research in that area). Now though I think there may be a simpler answer. > That was a HUGE breakthrough to move from concentrating on S itself, > where I had control variables I called n, alpha and k, and would > puzzle over things like what was the way to pick k, to figuring out > how to get p. And with the fundamental equations now known that completes the > foundation layer of surrogate factoring, and removes the need for > further brainstorming at this point, so there is no need for me to > discuss further. > I'm evaluating the situation this weekend as I ponder what to do next. It's a thorny problem with a lot of complexity introduced by the continuing refusal of the mathematical community to acknowledge this research. The simplest answer, if possible, may be to develop a working program and turn it over the United States Government. I would suggest that mathematicians around the world re-think their positions on this issue. My major concern is that I will end up giving up on my dream of restoring real mathematical research in number theory and leaping humanity forward mathematically, and instead find myself at the heart of military endeavors as my problem solving skills are turned instead to activities that are invisible to the world. If that happens none of your countries will have any chance of competing with the US within your lifetimes or the foreseeable future, and much of what I do will be invisible, but the impact will be unstoppable. James Harris === Subject: Re: JSH: Diminishing concerns on factoring posting-account=fwSgtAkAAACFnX70ssKwbvm9_oCZVHrx 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) For years I've had a major fear that if I had a major breakthrough > with integer factorization it might have a very bad, negative impact > on the global economy but now as I finish out the foundation layer of > that research it is clear that my research is not impacting the > world. æYeah, the world is having problems but those are its own. That was then and I still don't see how I'm impacting the world yet, > but my concerns have returned as I've solved the problem of finding > the odd prime p. > Without a real demonstration the only assumption that can reasonably be made is your claim Has No Substance To It!!!!!! === Subject: Re: JSH: Diminishing concerns on factoring posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) > For years I've had a major fear that if I had a major breakthrough > with integer factorization it might have a very bad, negative impact > on the global economy but now as I finish out the foundation layer of > that research it is clear that my research is not impacting the > world. æYeah, the world is having problems but those are its own. That was then and I still don't see how I'm impacting the world yet, > but my concerns have returned as I've solved the problem of finding > the odd prime p. Without a real demonstration the only assumption that can reasonably > be made is your claim Has No Substance To It!!!!!! Why? What would a demonstration prove? JSH === Subject: Re: JSH: Diminishing concerns on factoring SV1),gzip(gfe),gzip(gfe) For years I've had a major fear that if I had a major breakthrough > with integer factorization it might have a very bad, negative impact > on the global economy but now as I finish out the foundation layer of > that research it is clear that my research is not impacting the > world. æYeah, the world is having problems but those are its own. That was then and I still don't see how I'm impacting the world yet, > but my concerns have returned as I've solved the problem of finding > the odd prime p. Without a real demonstration the only assumption that can reasonably > be made is your claim Has No Substance To It!!!!!! Why? æWhat would a demonstration prove? JSH- Hide quoted text - - Show quoted text - It would prove that you have something worthwhile. === Subject: Re: JSH: Diminishing concerns on factoring posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) > For years I've had a major fear that if I had a major breakthrough > with integer factorization it might have a very bad, negative impact > on the global economy but now as I finish out the foundation layer of > that research it is clear that my research is not impacting the > world. æYeah, the world is having problems but those are its own. That was then and I still don't see how I'm impacting the world yet, > but my concerns have returned as I've solved the problem of finding > the odd prime p. Without a real demonstration the only assumption that can reasonably > be made is your claim Has No Substance To It!!!!!! Why? æWhat would a demonstration prove? JSH- Hide quoted text - - Show quoted text - It would prove that you have something worthwhile. So what if I don't demonstrate with a factorization of a very large number? What does that prove? JSH === Subject: Re: JSH: Diminishing concerns on factoring posting-account=aLpfCwoAAACh4BOs3HOlQBCoxUpEgyxc Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) Without a real demonstration the only assumption that can reasonably > be made is your claim Has No Substance To It!!!!!! Why? What would a demonstration prove? It would prove that you have something worthwhile. So what if I don't demonstrate with a factorization of a very large > number? What does that prove? It proves that you don't learn from your mistakes. === Subject: Re: JSH: Diminishing concerns on factoring posting-account=n1ZfDgkAAABbCs44qOtz8dP-RkWuEBif Gecko/2008052906 Firefox/3.0,gzip(gfe),gzip(gfe) > Without a real demonstration the only assumption that can reasonably > be made is your claim Has No Substance To It!!!!!! Why? æWhat would a demonstration prove? It would prove that you have something worthwhile. So what if I don't demonstrate with a factorization of a very large > number? æWhat does that prove? It proves that you don't learn from your mistakes. Yawn. And you do? If so, why are you replying to me now? If I could demonstrate with a large factorization I would have done it. I can't. James Harris === Subject: Re: JSH: Diminishing concerns on factoring It proves that you don't learn from your mistakes. >Yawn. And you do? If so, why are you replying to me now? >If I could demonstrate with a large factorization I would have done >it. >I can't. >James Harris Your admission proves you are only a troll, a one trick phony. === Subject: Re: JSH: Diminishing concerns on factoring posting-account=JpxxPAgAAAAgwzQIYqn4j6syK-YhOmcF Gecko/20071127 Firefox/2.0.0.11,gzip(gfe),gzip(gfe) > The simplest answer, if possible, may be to develop a working program > and turn it over the United States Government. > what a great idea! may I recommend to develop the following program: print Hello world! The impact of such program on the world will be truly unstoppable. === Subject: Re: JSH: Diminishing concerns on factoring > The simplest answer, if possible, may be to develop a working program > and turn it over the United States Government. what a great idea! may I recommend to develop the following program: print Hello world! The impact of such program on the world will be truly unstoppable. I think not. The current US Administration might well see such a program as dangerously close to expressing support for the United Nations. A program that printed Hello America! would be more likely to be accepted. rossum === Subject: Re: JSH: Diminishing concerns on factoring posting-account=OKTeIQkAAAAZk6JK1hK7-grwpoUDNy98 CLR 2.0.50727),gzip(gfe),gzip(gfe) > The simplest answer, if possible, may be to develop a working program > and turn it over the United States Government. what a great idea! may I recommend to develop the following program: print Hello world! The impact of such program on the world will be truly unstoppable. I think not. æThe current US Administration might well see such a > program as dangerously close to expressing support for the United > Nations. æA program that printed Hello America! would be more likely > to be accepted. That would include gays and immigrants. Perhaps print Hello Whute Anglo-Saxon Protestant Heterosexual Republicans! might have a chance. rossum === Subject: Re: Diminishing Returns in Game Engineering AngleWyrm: > I've added a section so that it is quite clear why, called > Double-Implementation > http://home.comcast.net/~anglewyrm/diminishing_returns.htm I don't think your argumentation is correct. What you have called the top (repeating) implementation is indeed used to create this rare/common disbalance, to make outstanding things rare. A great example is Battle Isle, where units' skill level grew slower as they became more experienced. You can come to this conclusion if you assume experience is proportional to the time spent in action and that achieving each next skill level requires more experience that the previous one. If you're twice more experienced than I, you're only one step more skilled (not twice as skilled). This is kind of a logarithm, Skill = log(Experience). And know what? This has been supported by the best judge -- the reality: WWII stats show that newbie soldiers who survived their very first battle had 5-10 times higher chence to survive in the future than complete newbies. So, the growth of Skill is fast in the beginning and slows down considerably at higher levels. Logarithm behaves just like this! Now, back to the number-of-stats problem. You say increasing this number does just the same. But using your assumtion about multiplication of stats (which I don't find correct...) I have come to an opposite conclusion. Let's assume the stats grow linearly (without loss of generality, as long as they're sufficiently correlated and therefore change at approximately the same rate, as you assume): x = 0.1, x^3 = 0.001 x = 0.2, x^3 = 0.008 (+0.007) x = 0.3, x^3 = 0.027 (+0.019) ... x = 1, x^3 = 1; x = 2, x^3 = 8; (+7) x = 3, x^3 = 27; (+19) As you see, the derivative only increases, meaning that the more outstanding characteristics will be even more likely than modest ones, which is actually disgusting to nature. Is it possible that your stats improve quicker and quicker with time, without a limit? This prevents the model from converging and makes it unstable: initially a little bit more skilled units will quickly develop so emormously an advantage over those that initially were just a tiny little bit weakier, that less and less units will make sence (0.0001 against 10000 is nothing, right). And will be a self-accelerating process, so the game will be reduced to race for experience, and the winner will be known quite early... As for scaling that you say doesn't effect things, you're again not right. Number one is the turning point for the X^N function. Plot the same graphs for values from 0 to 50 and see the difference ;) ... If we stick to the scale [0,1] -- things won't be any better: Near pont point one the skill gaining speed makes a leap and significally increases. Why? In reality I don't see no reason for it, so such a model would be at least unrealisic, and at most -- not playable. Anton === Subject: Wendy Bouchoux's answers to problems from Linear Algebra 18.700-MITOCW Attached are the answers to some of the problems from MITOCW Linear Algebra 18.700. begin 666 Linear Algebra 18.700-MITOCW-Wendy Bouchoux-Answer to Problem 1 in Problem Set 3.rtf