INVITED SPEAKERS:

Peter Andrews (CMU)
Tomek Bartoszynski (Univ. of Idaho)
James E Baumgartner  (Dartmouth)
Steve Buechler (Notre Dame)
Ambar Chowdhury  (UCon)
Randall Dougherty (OSU)
Ward Henson (Illinois at Urbana)
Roman Kossak (CUNY)
Andrzej Roslanowski (Hebrew Univ.)
Mati Rubin (U. of Colorado)
Rick Statman (CMU)

 At the end of this announcement find a list of titles and abstracts.

SUPPORT FOR GRADUATE STUDENTS:
We have limited funds to support grad students.  The cost of meals and
mileage will be reimbursed for graduate students attending (this means
for graduate students who use their own cars, not for graduate students
who are given a ride).  Other  requests for reimbursement by graduate
students will be considered on an individual basis.  Since funds are
limited, graduate students who anticipate significant mileage costs
should contact the organizer in advance.

The meeting will take place at Wean Hall on Saturday and Sunday  (the
third weekend of March).  Due to the large number of speakers
the meeting will run from 10:00 A.M. Saturday to 4 P.M. Sunday.  

In addition Ward Henson will give a Mathematics Colloquium on Friday
afternoon (April 2nd  4:30 P.M. - refreshments at 4:00 P.M.!).  You are
all invited.  

Carnegie Mellon Campus is on 5000 Forbes Ave.  in an area of Pittsburgh
called Oakland (about 3-4 miles east of downtown).

ACCOMODATIONS:
(1)   I have reserved a block of rooms at the nicest hotel which is
within walking distance from CMU  (it is also the least expensive!).
You should make the reservation yourself.  When making your reservations
please mention that you are part of the CMU logic meeting.  The cost is
$70 for single room.  The rooms will be reserved until 28th of February.
 You are encouraged to make the reservation early.

The hotel is:  "University Club"  located at  123 University Place (off
5th Ave.).  The telephone number is -  412- 621-1890.  This is a private
club (not a faculty club),  they have a dress code which is enforced
only in the dining room.

(2)  Also there is a  "Holiday-Inn" within walking distance from campus.
It is located on 100 Lytton Ave  (off 5th Ave.)  Telephone-
412-682-6200.  Single rooms start at $99  (the usual cost is  $124).

(3)  The "Hampton Inn" is also within walking distance.  It is on 3315
Hamlet St. (which is 2 blocks south to 5Ave.  off Craft street).  The
cost is $70 for single room.  The number to call is  412-681-1000.

(4)  In case you are driving and/or  you are interested in something
cheaper, I suggest staying at the  "Palace Inn"  which is located 10
miles east to CMU in Monroeville.  The charge for a single room is
$41.95.  The hotel is located on route 48 south, 100 feet from the
intersection with route 22  (which is about a quarter of a mile from the
Pittsburgh exit from the PA-Turnpike  - exit #6).

DRIVING INSTRUCTIONS:

LIST OF TALKS&ABSTRACTS:
-----------------------------
(1)  Speaker: Peter B. Andrews (CMU)
     Title:  Proving Theorems of Type Theory with TPS
     Abstract:  TPS is a theorem proving system for type theory (Church's
typed lambda-calculus) which has been under development for a number
of years.  TPS is a general research tool for manipulating wffs of
first- and higher-order logic, and for proving such wffs
automatically, interactively, or in a combination of these modes.  The
work has been done in collaboration with Matthew Bishop, Sunil Issar,
Dan Nesmith, Frank Pfenning, and Hongwei Xi.
        When proving a theorem automatically, TPS first searches for
an expansion proof of the theorem, using heuristics governed by flags
set by the user. It then translates this expansion proof into a
natural deduction proof.  An expansion proof is a structure which
represents the theorem, the terms with which quantifiers are
instantiated, and a tautology which is essentially a pruned Herbrand
expansion of the theorem.  It contains in a nonredundant way the basic
combinatorial information required to construct proofs of the theorem
in many styles.
        Examples of theorems which TPS can prove automatically are
given to illustrate certain problems of theorem proving in
higher-order logic and aspects of TPS's behavior.  TPS can prove
certain theorems which require instantiating quantifiers on
higher-order variables with expressions containing connectives and
quantifiers. We show how TPS finds a proof for a theorem which asserts
that if some iterate of a function f has a unique fixed point, then f
has a fixed point.

(2)  Speaker:  Tomek Bartoszynski (Univ. of Idaho)
     Title: Sets of reals related to cardinal invariants.
     Abstract:  TBA

(3)  Speaker: James E. Baumgartner (Dartmouth)
     Title: Large cardinals in polarized partition relations
     Abstract:  TBA

(4)  Speaker:  Steve Buechler (Notre Dame)
     Title: Issues related to Vaught's conjecture
     Abstract: With the proof of Vaught's conjecture for superstable
theories of finite rank attention naturally turns to general
superstable theories and beyond.  In this talk I will first
discuss the methods that went into the proof of the finite rank
case and then ideas for the extension of these methods.  In
particular I will introduce an alternative to the standard
regular type technology which may enable rank and dimension-like
arguments even in arbitrary stable theories.  This alternative
notion of rank appears to be more natural in a study of Vaught's
conjecture than currently established notions of dependence.

(5)  Speaker:  Ambar Chowdhury  (UConn)
     Title: Smoothly approximable structures and the small index property.
     Abstract: An omega-categorical structure, M, is smoothly approximable
if M is the union of an increasing chain of finite homogeneous substructures.
Cherlin and Hrushovski have shown (amongst other things) that any such
structure is built in a certain way from `classical geometries'. I will
report on recent work with B. Hart and Z. Sokolovic, aimed at showing that
a smoothly approximable structure, M, has the small index property; i.e.
every subgroup of Aut(M) which has index less than the continuum, is
open in Aut(M).

(6)  Speaker: Randall Dougherty (OSU)
     Title:  Left-distributive algebras and critical points of
elementary embeddings
     Abstract:  TBA

(7)  Speaker:C. Ward Henson (Illinois-Urbana)
     Title: Model theory for Banach space structures
     Abstract:  The main theme of this talk is to explain how model theory
can be made an effective tool for handling structures which come from
functional analysis, topology, geometry, etc.  By "model theory" we
mean not just the use of saturated models, but also the use of more
delicate aspects of model theory such as the omitting types theorem,
quantifier elimination, stability theory, etc.  The main emphasis will
be on omitting types, and this will provide logicians with the technical
ideas that are behind what Henson will discuss in his Math Colloquium on
Friday.

(8)  Speaker: Roman Kossak (CUNY)
     Title: Arithmetic Saturation
     Abstract: Arithmetic saturation is a natural variation on
the notion of recursive saturation.
It has proven to be useful in the study
of countable recursively saturated models of PA and their
automorphism groups. For example, if M is a countable recursively
saturated model of PA, then the following are equivalent:
(i) M is arithmetically saturated;
(ii) The standard system of M forms a model of ACA_0;
(iii) The automorphism group of M is not the union of a
countable chain of its proper subgroups.
I will discuss these and other results from two papers
of Jim Schmerl and myself: "The automorphism group of
an arithmetically saturated model of Peano Arithmetic",
to appear in J. London Math. Soc., and "Arithmetically
saturated models of arithmetic", to appear.

(9)  Speaker: Andrzej Roslanowski (HUJI)
     Title:     Cardinal invariants of Boolean algebras and ultraproducts:
  s (spread), hd (hereditary density) and hL (hereditary Lindelof degree).
     Abstract:  This is a joint work with Saharon Shelah. We construct
Boolean algebras $B_n$ such that for every non principal ultrafilter $D$
on $\omega$
$inv(\prod_{n\in\omega} B_n/D) > \prod_{n\in\omega} inv(B_n)/D$
where inv is one of the following cardinal invariants of Boolean
algebras: s,hd, hL. The construction involves some very basic elements
of pcf-theory and answers three of the problems of Monk.

(10)  Speaker:  Mati Rubin  (Ben Gurion, currently at Univ. of Colorado)
     Title:  Locally moving groups and reconstruction problems.
     Abstract:  TBA

SHA=25:
275B3 570L3 570L4 870I3 870I4

SHA=49:
546F2 858K2

The numbers refer to his table in _Algorithms for Modular Elliptic Curves_.
For instance 275B3 is the curve  y^2 + a1 x y + a3 y = x^3 + a2 x^2 + a4 x + a6
where [a1,a2,a3,a4,a6]=[0,1,1,-195508,-33338481], a twist of one of the curves
of conductor 11; and 546F2 has [a1,a2,a3,a4,a6]=[1,0,0,-3674496,-2711401518].

These curves were originally found by using the conjectural analytic formula
for Sha of Birch and Swinnerton-Dyer, but Dick Gross soon observed that
the fact that SHA has 5- or 7-torsion in those cases follows from Cassels'
theorem that the BSD Conjecture is compatible with isogenies: each of
these curves has a 5- or 7-isogenous curve the product of whose c_p's
is 5^2 or 7^2 times smaller.

To see how I went about computing these, see my paper on "Heegner
point computations" in the proceedings of the first international
symposium on algorithmic number theory (ANTS-I), edited by Adleman
and Huang (Lecture Notes in Computer Science #877).

I have two questions pertaining to the history of algebra:

1. I am interested in biographical details, perhaps a bibliography,
and other salient information, about T. Molien.

2. I would like to know the history of the Kronecker product.

On a sphere, draw four meridians connecting the north pole to
the south pole; label them C,D,E,F proceeding eastward.  Also
draw a latitude circle for each integer, where higher-numbered
circles are farther north, with circle number  n  tending to the
north pole as  n->infinity  and to the south pole as  n->-infinity.
This divides the sphere up into a grid of quadrangles; add
a southeast-to-northwest diagonal to each quadrangle to make
a triangular grid.

This triangular grid can be drawn using eight paths:
   A_1 =  ... -C0-C1-C2- ...
   A_2 =  ... -C0-D0-C1-D1-C2-D2- ...
   A_3,...,A_8 = the same things shifted east by one, two, or
                 three meridians.
Or it can be drawn using four paths:
   B_1 =  ... -C0-D0-E0-D1-E1-F1-E2-F2-C2-F3-C3-D3-C4-D4-E4- ...
   B_2 =  ... -C0-C1-D0-D1-D2-E1-E2-E3-F2-F3-F4-C3-C4-C5-D4- ...
   B_3,B_4 = the same things shifted east by two meridians.
Now let B_5,...,B_12 be the two poles listed four times each
to complete the counterexample.  (Each grid intersection other
than the poles is in three of the A's and three of the B's.)

By puncturing the sphere at a point not on the grid, one can turn
this into a counterexample in the plane.

This reduces the question to dimension 1, where it is almost immediate.

Let's do one dimension first.  Let A_i, i in I, be a finite indexed
family of nonempty compact convex subsets of R.  In this case the A_i
are finite closed intervals [a,b].  Let f(x) = the cardinality of the
set {i in I | x in A_i}.  For any x, for sufficiently small positive
e, the quantity r(x) = f(x) - f(x+e) is the cardinality of the set
{i in I | x is the right endpoint of A_i}, and l(x) = f(x) - f(x-e) is
the cardinality of the set {i in I | x is the left endpoint of A_i}.
Note l(x) and r(x) are zero outside a finite set, since there are only
finitely many intervals and finitely many endpoints.  Then

   |I| = sum over x of l(x) = sum over x of r(x),

since each interval has exactly one left and one right endpoint.
We're done, since these quantities depended only on f.

Note that if some of the A_i can be empty, then this calculation gives
you not |I| but the cardinality of the set {i in I | A_i is nonempty}.
We'll use this in the inductive step below.

Let's do 2 dimensions for illustration.  Let A_i, i in I, be a finite
indexed family of nonempty compact convex subsets of R^2.  Let f(x,y)
be the cardinality of the set {i in I | (x,y) in A_i}.  Let p:R^2 -->
R be the projection onto the second component.  The image

   p(A_i) = {y | there exists x such that (x,y) in A_i}

is nonempty, compact and convex, therefore a nonempty closed interval
in R.  Consider also the set

   A_i(y) = {x | (x,y) in A_i}.

This is a compact convex subset of R, hence either empty or a closed
interval.  It is nonempty iff the line in R^2 through (0,y) parallel
to the x axis intersects A_i.  Now define

f'(y) = the cardinality of {i in I | y in p(A_i)}               (*)
      = the cardinality of {i in I | there exists x s.t. (x,y) in A_i}
      = the cardinality of {i in I | A_i(y) is nonempty}        (**)

By the one-dimensional argument using characterization (**), f' is
determined by f.  By the one dimensional argument using
characterization (*), |I| is determined by f'.  Therefore |I| is
determined by f.

For arbitrary dimension d >= 2, let A_i, i in I, be a finite indexed
family of nonempty compact convex subsets of R^d.  Let f(x_1,...,x_d)
be the cardinality of the set {i in I | (x_1,...,x_d) in A_i}.
Consider the projection p along the x_1 axis.  The projected image

p(A_i) = {(x_2,...,x_d) | there exists x_1 s.t. (x_1,...,x_d) in A_i}

Let $H_r^2(T)$ be real $H^2$, i.e., the collection of real parts of $H^2(T)$
functions, with norm inherited from $H^2(T)$ (i.e., norm given by adding in
$i$ times the conjugate function, and taking the $H^2$ norm).
Then, $H_r^2(T)$ precisely coincides with $L_r^2(T)$ (real $L^2$), at least
set-wise.  What about the norms?  Write $f \in H^2(T)$ as
$$
        \sum_{n=0}^\infty a_n z^n + i \sum_{n=0}^\infty b_n z^n,
$$
where the $a_n$ and $b_n$ are real.  Since our question is invariant under
adding an imaginary constant, assume $b_0=0$.
$\| f \|_{H^2}^2=\sum_{n=0}^\oo (a_n^2+b_n^2)$.  Now,
$$
        \Re f = \sum_{n=0}^\infty a_n \cos n\theta - \sum_{n=0}^\infty
                b_n \sin n\theta.
$$
It is easy to verify that square of the $L^2$ norm of this is
$$
        a_0^2+(1/2)\sum_{n=1}^\oo a_n^2+b_n^2
$$
Thus,
$$
        \| \Re f \|_2^2=(1/2)( \| f \|_{H^2}^2 + a_0^2 ),
$$
providing $\Im f(0)=0$.
Therefore,
$$
        \| f \|_{H^2}^2=2\| \Re f \|_2^2-a_0^2.
$$

This is the functional we need to worry about.  In other words, given a
real $L^2(T)$ function $g$, we must find a nonnegative $h$ such that
$$
        d(g,h)=2\| g-h \|_2^2-(\int_T (g-h))^2
$$
is minimal.  This seems to be a bit easier to handle as it contains no
references to analytic functions, and just concerns real ones.  If we didn't
subtract off the second term, then the answer to your problem would be trivial.
It suffices to consider the conjecture (that $h=g^+$ works) for simple
functions $g$.  I do not know whether in such a case the optimal $h$ (assuming
such exists) is constant on the same sets on which $g$ is constant.  If we
knew that a unique optimum exists (and the poster hinted that he knew this),
then we would know that $h$ is constant on the sets on which $g$ is constant.

In any case, we can see that the conjecture is false.  For if it is true, then
ccertainly we can restrict our attention to those $h$ which are constant
on the same sets on which $g$ is constant.  Suppose $T=A_1 \cup ...
\cup A_n$ where $A_1,...,A_n$ are sets of measures $a_1,...,a_n$, respectively.
Let $g$ take the values $t_1,...,t_n$ on $A_1,...,A_n$, respectively, and
let $h$ take the values $u_1,...,u_n$.  Then
$$
        d(g,h)=2( \sum_{j=1}^n a_j (t_j-u_j)^2 )-(\sum_{j=1}^n a_j (t_j-u_j) )^2.
$$
In principle, the question is now a simple optimization problem with
the constraint that the $u_j$ are nonnegative.  One could probably proceed
now by using standard multivariate calculus.
$$
\partial d(g,h) / \partial u_j = 4a_j (u_j-t_j)
                        - 2a_j (\sum_{i=1}^n a_i (u_i - t_i)).
$$
At the extremum, this must vanish for every $j$ such that $u_j \ne 0$.
Thus, in principle, it should be possible to work out the optimum for any
$n$ by taking all possible subsets $J$ of $\{ 1,...,n \}$, setting
$u_j=0$ for $j\in J$, then solving the resulting equations for the
remaining $u_i$---which can be done quite easily since the equations
are all linear---and then checking for which of the subsets $J$ this is
largest.  To see that the conjecture is false, suppose that $u_i=t_i^+$.
Then the second term in the partial derivative formula above is negative
(strictly negative, in fact, if $g$ is not nonnegative).  Then, suppose that
$t_j > 0$ for some $j$.  For this $j$ we easily see that the partial
derivative given above is strictly negative since $u_j=t_j$.  Hence the
choice $u_i=t_i^+$ is not even a local extremum, much less a global
minimum.  Hence, assuming the above is correct, the conjecture is false.

If someone can actually solve the optimization problem above in a nice
way, I would be grateful for being told of the solution.

I have not worked out what happens in the case where all the $t_j$
are nonpositive.  I think that under those circumstances we can find
some choices of $t_j$ and $a_j$ for which it the conjecture is false
and some for which it is true.

Let n be a positive integer, and let r_i for i in 1,...,n-1 be a
real number in [0,1].  For each i between 1 and n define a vector v_i
of length n by (r_{i-1}, ... r_1, 1, 0, ... ,0).

These vectors define a parallelepiped P.  The set of vertices Q
is given by Q = \sum{i=1}^n a_i v_i where a_i is in {0,1}.
P is given by \sum{i=1}^n c_i v_i where c_i is in [0,1].

Let k be a positive real number.  We want to center cubes of radius k
at each vertex in Q, and we ask whether the union of these cubes covers
P.  In symbols, if K = [-k,k]^n, is P in Q+K?

Questions:
a) For a specific n and choice of r_i, what k is necessary to cover
   P by Q+K?  
b) Increase the dimension of the problem by increasing n and taking
   r_i = 0 for i greater than the original n.  What k is necessary
   as n -> infinity?
c) For which choices of r_i is the acceptable k less than 1, and how
   much less?

Any leads would be appreciated,

Could anyone tell me what the numerical continuation method is about ?
I would also really appreciate any examples where the method could be used
and any references for the method.  Thanks for yout time.

        Hello,

Implementing some algorithms, I use a basis of orthogonal complex
polynomials using a scalar product of the form :

                      /          ----
          < f , g > = |   f(z) * g(z) * w(z) * dz
                      /
                       gamma

where "gamma" is complex curve.

I would like to know if the common 3-terms recurence for real orthogonal
polynomial holds for the complex case too.
                         -------

If not ... would you have any references or counterexamples on this
subject ?

Thank you very much for your help.

                                        Vincent

--

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  Vincent HEUVELINE
      IRISA
  Campus universitaire de Beaulieu
  35042 RENNES Cedex (FRANCE)

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