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SOME NEW RESULTS IN THE FIELD OF DISCRETE MATH/DESIGNS/CODES
Date  : 96/11/27 We
Author: Uenal Mutlu (bm373592@muenchen.org)
The following bounds of enumerated 'm=t COVERING DESIGNS' are to
the best of my knowledge *NEW* upper bound results compared to
known published or online available results (cf. [1],[2],[4]).
Definition:
 A Covering Design C(v,k,t,m,l,b) is a pair (V,B), where V is a set
 of v elements (called points) and B is a collection of b k-subsets
 of V (called blocks), such that every m-subset of V intersects at
 least l members of B in at least t points (v >= k >= t and m >= t).
New upper bounds for C(v,k,t,m,l=1,b). Listed are m=t designs only:
 k t m	v     b (bOld)	Remarks/Author/Method
------------------------------------------------------------------
 6 4 4 15   118 (120)	Rade Belic, [3]
 6 5 5 14   377 (378)	Rade Belic, [3]
 6 5 5 15   609 (610)	Combine-Construction [1]
 7 5 5 17   408 (463)	Rade Belic, [3]
 7 5 5 18   618 (663)	Combine-Construction [1]
 7 5 5 19   772 (824)	Turan Theory/Sidorenko Construction [1]
 7 5 5 20  1115 (1153)	Combine-Construction [1]
 7 5 5 21  1424 (1476)	[3]
 7 5 5 22  1938 (1990)	[3]
 7 5 5 23  2405 (2458)	[3]
 7 5 5 24  3096 (3208)	[3]
 7 5 5 25  3772 (3830)	[3]
(bOld refers to the upper bounds of [2] on 96/11/26)
The new upper bounds were obtained by some published (see [1]) and
some not yet published algorithms and programs of Rade Belic [3]
and Uenal Mutlu (including implementations of some recursive
constructions from [1]).
These results will also improve some other designs with higher
k,t,m if at least some of the well known recursive construction
methods are applied.
Not shown are the latest improvements and new upper bounds for
designs with m > t. A list containing all upper bounds of enumerated
designs in the range k <= 7, v <= 54, t <= 7, m <= 7 and b <= 9999
can be requested from the author [5]).
Pointers to further improvements and new results are welcome.
Uenal Mutlu
References:
[1] D.M.Gordon, O.Patashnik, G.Kuperberg "New Constructions for
    Covering Designs", J.Combin.Designs 3.4 (1995) 269-284
    (for updates and errata see [2])
[2] Dan Gordon's "La Jolla Covering Repository" at
    http://sdcc12.ucsd.edu/~xm3dg/cover.html
    (as of 96/11/26)
[3] SOPT v1.0 07/96 (96/11/19) - Design Optimization Program
    written by Rade Belic
[4] D.R.Stinson "Coverings" in C.J.Colbourn, J.H.Dinitz (eds.)
    "The CRC Handbook of Combinatorics" CRC Press (1996) 260-265
[5] Uenal Mutlu "List of Covering Designs" (enumerated upper bounds)
    Includes also m > t designs. Current list is LST1127A.ZIP
-- Uenal Mutlu (bm373592@muenchen.org)   
   Math Research, Designs/Codes, Data Compression Algorithms, C/C++
   Loc  : Istanbul/Turkey + Munich/Germany

I am looking for multivariate analogues of Euler-MacLaurin's
quadrature formula. Has anyone ever come across something
of that kind ?
I would appreciate any hints, references etc.
Hartmut Fuehr

 "A countable compact Hausdorff space has an isolated point."
Is AC required to prove this?  If the second axiom of countability
(or even more, metrizability) are postulated, then we can prove it
in ZF, but what about the statement given?
-- 
Gerald A. Edgar                   edgar@math.ohio-state.edu

Peter Flor  wrote concerning my question 
whether there exists a homeomorphism between the Cantor set and a 
subset of the unit interval with measure one 
 () If S is a closed subset of [0,1] having measure one, then the 
 () complement has measure zero. Being open, it must be empty. So 
 () the answer to your question is " no". 
This does not answer the question. It is not asked for a _closed_ 
subset of the unit interval, but for a Borel (or a Lebesgue) set. 
I find it plausible, though, that `no' is the right answer. 
-- 
		Hans Crauel	crauel@saar.de
		Hellwigstr. 17, 66121 Saarbruecken
		+49-681-64516 

I have been looking at the properties of a family of polynomials
generated by the recurrence relation
               P(n+1) = xP(n) - P(n-1)
               P(0) = 0,   P(1) = 1
They have many interesting properties; e.g., P(n) has n-1 real
roots, all contained in the interval [-2,2], with the apparent
limit for n going to infinity of the largest root being 2.
This came up accidentally in some work on fractal generation,
and is an area of math I have not studied.  Any references for
recurrence relations defining polynomial families, or even on
this particular one appreciated.
BV

In article <57ga5l$8rk@charity.ucr.edu>, John Baez  wrote:
>The only integer solution of
>
>           1^2 + 2^2 + ... + n^2 = m^2,
>
>is the solution n = 24, not counting silly solutions like n = 0 and
>n = 1.
>It seems the Lucas didn't have a proof of this; the first proof is
>due to G. N. Watson in 1918, using hyperelliptic functions.   
I got this information from Jet Wimp's review, and haven't actually
seen the proof, but David Morrison pointed out in email that since
the sum on the left is n(n+1)(2n+1)/6, this problem can be solved by
finding all the rational points (n,m) on the elliptic curve 
         (1/3) n^3 + (1/2) n^2 + (1/6) n = m^2
which is the sort of thing folks know how to do.  For a wee bit on 
elliptic curves, see "week13".
Also, Robin Chapman pointed out that Anglin's elementary proof
also appears in the American Mathematical Monthly, February 1990,
pp. 120-124, and that another elementary proof has since appeared
in the Journal of Number Theory.
>In dimension 16 there are only *two* even unimodular lattices.  One
>is E8 + E8.  A vector in this is just a pair of vectors in E8.  The
>other is called D16*, which we get the same way as we got E8: we
>take a checkerboard lattice in 16 dimensions and stick in extra spheres
>in all the holes.  More mathematically, to get E8 or D16*, we take all 
>vectors in R^8 or R^16, respectively, whose coordinates are either 
>*all* integers or *all* half-integers, for which the coordinates add 
>up to an even integer.  (A "half-integer" is an integer plus 1/2.)  
I was being very silly in my notation here; that other even unimodular 
lattice in 16 dimensions is called D16+, not D16*, and it is definitely
not the dual of D16.   
In fact, we can play this trick in any dimension.  We start with 
the checkerboard lattice Dn in dimension n --- this is the lattice 
consisting of all integer-coordinate points whose coordinates 
add up to an even integer --- and double its density by throwing
in another copy of Dn shifted over by the vector (1/2,...,1/2). 
Conway and Sloan call the result Dn+.  It's only a lattice when
n is even, but when n = 3 it's the pattern that carbon atoms form
in a diamond!  It's only when we get up n = 8 that we can start
with tightly packed spheres centered at the points of Dn, 
and put in new spheres centered at the other points of Dn+ that are 
just as big, without having the spheres overlap.  The result being
E8, perhaps we could call E8 an "eight-dimensional diamond", which
at least conveys some of its crystalline beauty.  D16+ is then the 
"sixteen-dimensional diamond".  
>On the other hand, the 
>group SO(32) gives us the D16* lattice --- or at least some very
>related lattice; I always get confused about this point, and I'm too 
>tired to figure it out now, but perhaps some kind reader will
>confirm or correct me here.  
Well, it seems that D16+ is neither the weight lattice nor the 
root lattice of SO(32) or Spin(32) or anything quite like that.  
In his review article on the heterotic string David Gross says 
it's the lattice generated by the weights of the adjoint rep of 
SO(32) together with one of the two spinor reps.  
>Well, in dimension 24, there are *24* even unimodular lattices,
>which were classified by Niemeier.  A few of these are obvious, like 
>E8 + E8 + E8 and E8 + D16*, but the coolest one is the "Leech 
>lattice", which is the only one having no vectors of length 2.  
>This is related to a whole WORLD of bizarre and perversely fascinating
>mathematics, like the "Monster group", the largest finite simple 
>group --- and also to string theory.  
Joshua Burton caught me here; I meant to say the largest *sporadic*
simple group.   This is especially galling because I had been 
preening myself ever since catching someone else who made this 
error on sci.math.research a while back.
It's rather spooky, isn't it, how there are exactly 24 even unimodular
lattices in 24 dimensions?  I'll probably say a bit more about that
one of these days, but for now let me just answer the obvious question:
what about in 32 dimensions?  Well, Conway and Sloane's book does a
nice calculation that shows there are more than 80 MILLION inequivalent
even modular lattices in 32 dimensions!  

I'm interested in non-Hausdorff topological spaces (other than the trivial one)
which have some mathematical significance, since most topological works seem
to confine themselves to considerations of Hausdorff spaces, and their
properties.   Are there any standard interesting non-Hausdorff spaces?
Alan Bain

	In article <57i33b$ac7@coli-gate.coli.uni-sb.de>,
			Hans Crauel  wrote:
> Peter Flor  wrote concerning my question 
> whether there exists a homeomorphism between the Cantor set and a 
> subset of the unit interval with measure one 
>
>  () If S is a closed subset of [0,1] having measure one, then the 
>  () complement has measure zero. Being open, it must be empty. So 
>  () the answer to your question is " no". 
>
> This does not answer the question. It is not asked for a _closed_ 
> subset of the unit interval, but for a Borel (or a Lebesgue) set. 
> I find it plausible, though, that `no' is the right answer. 
	If a subset S of [0, 1] is homeomorphic to the Cantor set then
S is compact, since the Cantor set is compact.  Compact subsets of [0,1]
are always closed.  Therefore if the set you are seeking exists then it
is closed.
	Mike Hardy
Michael Hardy
hardy@stat.umn.edu

Alan Bain (alanb@chiark.greenend.org.uk) wrote:
: I'm interested in non-Hausdorff topological spaces (other than the trivial one)
: which have some mathematical significance, since most topological works seem
: to confine themselves to considerations of Hausdorff spaces, and their
: properties.   Are there any standard interesting non-Hausdorff spaces?
: Alan Bain
Algebraic varieties with their Zariski topology are not Hausdorff
and, more generally so are schemes (except in trivial cases).
Felipe

8007@BayLine.Com (lisa) wrote:
>I am after a simple version of Ito's lemma.  Does anyone have a reference.  
>It is to do with analysing Black-Scholes option pricing model.  I think
>that there may be a simple error there.
>
It is brave of you to think that you have found an error, either in Ito's 
lemma or in a complete proof of the Black and Schole theorem for pricing 
European options since these things have been done over and over again. 
But maybe!? (I am not sure which one you think has the error.) On the 
other hand, most so-called proofs of the B-S theorem that you find in 
finance books are not really proofs. But I would like to know what error 
you feel that you have spotted.
For an outline of a careful proof you might like to look at my article 
"Optional mathematics is not optional" in the September issue of the 
Notices of the American mathematical Society.
John