Here's something...(and how about that quote in the first signature!). I don't know what axaf is. - adam |The originator of this message is Dimitri Mihalas. | |For those who want the condensed version, the software that |Perkin-Elmer used to polish the mirror was faulty. | |Robert W. Spiker, UVa Dept. of Astronomy |-------------------------------+ It is truly written that a man has five |rws3n@astsun.astro.virginia.edu| times as many fingers as ears, but only | or @bessel.acc.virginia.edu | twice as many ears as noses. | | |Message follows: |---------------------------------------------------------------------- |in case you have not heard: from a reliable inside source i found out |that the problem with ST is that the SOFTWARE driving the polisher was |defective. the corrections for spherical aberration were put in with the |wrong sign. consequently the mirror is not corrected for sph. abb., but |has an added dose of it. | |the error was not detected during testing because no test with collimated |light was ever done. (editorial remark: unthinkable!) apparently this was |a $30M economy measure in the face of the Challenger accident. likewise |none of the optics were ever tested in vacuum. the primary was and is |"perfect" relative to the specified curve; but alas the specification |was wrong. sigh. | |>from my amateur astronomer days (does that include 1990?) i recall that |spherical aberration is EASY to detect with the foucault test, which is |done with a pinhole, not collimated light. it is hard to believe that |ANYONE could have made such a blunder.. | |the only reason that people know this much is that the same software |was used for AXAF. the errors there were so huge as to be immediately |noticeable, and when the software was corrected, the mirror was "perfect". |i don't know whether the information from axaf was available prior to |the launch of ST, but it seems that it had to be. in which case one |wonders why PE didn't issue a "hold everything!". | |the future: no chance of bringing the whole telescope down for a refit. |best plan is to design compensating optics into the lightpath for future |instruments: relatively easy to do. but that will still take 3-5 years. | |i suppose it's "win a few, lose a few..." but i personally think that |nasa, the government, and the people should stick it into PE and TURN |it hard until they agree to refund the cost of the mistake and of the repairs. |i'm sick of seeing defense and defense-related contractors get away |with bloody murder and just get fatter and fatter on the profits. | |back to theory |dimitri |--------------------------------------------------------------------------- From: INHB000 <INHB@MUSICB.MCGILL.CA> Dear Bob: This is, we all hope, the last communication on the subject of HSP theorems. I will probably be giving a talk on it in Montreal and will try to have a paper ready for distribution then. I will cast this in terms of posets, although it really all goes through for an arbitary locally presentable category and acessible theory. Let E/M be a factorization system on Pos such that maps in E are epi and in M are mono. At first I thought there were only the two extremal such factorizations, but now I think there are others. In any case, let it be chosen and fixed. Suppose now that T is an accessible triple on Pos. The Kleisli category K is then accessible, as is the free functor F:Pos -->K. It is folklore that the category of T algebras is equivalent to the category that has as objects pairs (G,b) where b is a poset and M:K* (that is the opposite of K) to Set such that GF=Hom(-,b). In fact, you don't have to include b as part of the structure, but it is convenient. Morphisms are natural transformations of functors (which induce unique morphisms between the representing objects. Let us say that (G,b) is an M/U-subobject of (G',b') if b-->b' is in M. By an HSP subcategory of the category C of T-algebras, I will mean a subcategory closed under U-split epis, M/U-subobjects and products. One way of getting an HSP subcategory is the following. By a Horn, I mean a diagram in K of the form Ff Fn----->Fm \ \ \ g \ \ \ v Fk such that f is in E. An algebra (M,b) satisfies this Horn if there is a factorization as indicated: MFf MFn<-----Fm ^ ^ \ | \ | Mg \ | \ | \ | \ | Fk The full subcategory of objects that satisfy a Horn or any class thereof is an HSP subcategory of the category of algebras. It would be nice if the converese were true, but it isn't. What happens is that you can iterate this construction and get a new HSP subcategory. In the right kind of example, the itereated construction is not describable as the objects that satisfy a Horn. I will give some examples later. The theorem is that every HSP subcategory is given as the intersection of a possibly transfinite chain of HSP category each derived from the preceding ones either as an intersection at limit ordinals or by satisfying a class of Horns at non-limit ordinals. And, of course, conversely. I still don't know how to characterize a single step HSP subcategory. That is probably still interesting. Nonetheless, this seems to be the best one cne can do in general. I also have a condition sufficient that a sequence of Horn constructions is a single one. A couple of examples. Consider a theory with one 2-ary operation we will call sigma such that sigma(x,x)=x. The free algebra generated by 2, which we will represent as x<y has just three elements x,y and sigma(x,y). Now form the Horn Ff F(1+1)---->F2 \ \ \ g \ \ \ v F2 where f:1+1-->2 is the inclusion. Here we are using the epi/regular mono factorization system. g takes the first generator of F(1+1) (which, as it happens is 1+1) to y and the second one to sigma(x,y). An algebras this Horn iff it satisfies x<y ==> y<sigma(x,y) In the full subcategory defined by this Horn, the free algebra generated by 2 is now infinite, since it now includes elements like sigma(x,sigma(x,y)) and sigma(y,sigma(x,y)). We could now take a Horn that forced these last two elements to be equal. (The top of the Horn is F(1+1)-->F1). The solutions of this Horn are still an HSP subcategory of the category of algebras. But it cannot be given by one step because the terms that were set equal weren't even elements in the first free algebras. Here is an example in which we use the regular epi/mono factorization system. Here we can use only equations, since the top of the Horns must be coequalizers, which means we are putting in equations. It might seem that in this case, Horns can be composed, but isn't so. Consider two operations phi and psi of type 1-->2. This means essentially that we have operations phi_0<phi_1 and psi_0<psi_1. Suppose in addition we have a 3-ary operation tau such that tau(x,x,y)=x and tau(x,y,y)=y. Then the free algebra generated by a single x has infinitely many elements like phi_0(x)<phi_1(x) and psi_0(phi_1(x))<psi_1(phi_1(x)), but no terms built from tau because there are no non-trivial 3 chains. Now add the equation phi_1=psi_0 and all of a suddent there are 3 chains such as phi_0<phi_1=psi_0<psi_1. Now we could add the equation x=tau(phi_0(x),phi_1(x),psi_1(x)). This equation cannot be stated in the first algebra because the necessary elements weren't there. Here is an interesting example. Begin with the theory of one omega-ary operation I will call lim. So to each sequence x_0<x_1<... there is an element lim x_i. Add the Horn that embodies the inequalities x_j<lim x_i for each j. This is done by a Horn whose top is F(1+1)-->F2 as with the case of a binary op above. The next Horn looks like Ff F(1+1)---->F2 \ \ \ g \ \ \ v F(omega+1) The map f is as before. For g, take the map that takes the first element to lim x_i and the other to x_{omega}. The Horn forces the condition lim x_i<x_{omega} and an algebra satisfies it, in addition to the ones above iff lim x_i is the least upper bound. The algebras are the omega-CPOs. There is nothing to prevent you from sticking to discrete theories, in which all the ops have discrete domain and domain. We can also make the ops be increasing in some variables and decreasing by looking at Horns of the form Ff F(1+1)---->F2 \ \ \ g \ \ \ v F(n2+m2) Here by n2, I mean the sum of n copies of 2 and similarly with m2. Thus n2+m2 is the sum of n+m arrows, say x_i<y_i for i=1,...,n+m. Given an operation sigma on n+m discrete elements, let g take the first element of 1+1 to sigma(x_1,...,x_n,y_{n+1},...,y_{n+m}) and the second element to sigma(y_1,...,y_n,x_{n+1},...,x_{n+m}). An algebra satisfies this Horn iff it is increasing in the first n variables and decreasing in the last m. Thus this analysis actually can be specialized to the question as first posed. Michael