I am having a hard time proving that
If kn(Tn-theta) tends in law to a limit distribution H which does not assign probability 1 to any single point, and if kn --> 0 (kn is a sequence), then Tn does not tend to theta in probability.

I know that because kn(Tn-theta) tends in law to a limit distribution H, kn(Tn-theta) is bounded in probability, meaning P{kn|Tn-theta|1-delta, for any delta, for some K, for large enough n.

I have tried to find a positive lower bound for P{|Tn-theta|
I would appreciate some help.

Thanks in advance.

Is there a linearly ordered field extending the reals (hopefully not
exceeding the cardinality of the set of the reals), in which every
increasing sequence has a limit?

I was thinking about hyperreals or Conway numbers but couldnt quickly
derive an answer. Hyperreals seem not have this property.

On Jan 25, 1:30 pm, Blake Winter gmail.com> wrote:

Consider a differentiable embedding f : S^1 x [-1,1] -> S^4.
Since it is not surjective, we may remove a point from S^4 not
in the image and get an embedding f : S^1 x [-1,1] -> R^4.
This does not change anything if we intend to untwist f.

Now the equatorial circle, i.e. the restriction of f to S^1 x {0},
is unknotted in R^4 (as all closed curves are) so it is
diffeotopic to a standardly embedded S^1 in R^4. Composing f
with the end of the diffeotopy, we see that we may assume that
f : S^1 x [-1,1] -> R^4 is a differentiable embedding such that
its restriction to S^1 x {0} is the map

sigma : t in S^1 --> (cos(t), sin(t), 0, 0) in R^4.

Now by `shrinking' f along the direction of [-1,1], using
the fact that f is an inmersion, and working a bit, we see
that f is diffeotopic to a map of the form

(t, s) in S^1 x [-1,1] --> sigma(t) + s v(t) (*)

Hi, I'm a bit confused about some of the theory of schur multipliers
on schatten classes. One definition that is given for schatten class
of B(l^2) is that if we identify matricies as operators in the
standard way, then then a matrix A is a schur multiplier on Sp
(B(l^2)) if A*B is in Sp (B(l^2)) (treating a matrix as an operator
wrt the standard basis of l^2) whenever B is a schatten p class
matrix, where * is pointwise (Schur) multiplication.

But more generally others define it for a metric space X with measure
('more generally' because if we use the standard basis for l^2,
consider these matricies as integral operators on l^2 with counting
measure, then schur multiplication corresponds to multiplication of
the kernels) as a function K on X x X is a schur multiplier on
Sp(B(L^2(X))) if whenever B is a function on X x X that induces a
schatten p class integral operator on L^2(X), then K B (ordinary
pointwise multiplication of kernels) induces a schatten p class
integral operator on L^2(X).

The University of T

If the sum over all positive and negative integers n and m of the quantity
A_m * cos( b*(n-m) ) * exp( -a^2 * (n-m)^2 )
is proportional to a^(-1/2), what, if anything, can I deduce about the
sequence A_m? a, b, and A_m are real. The sequence A_m is stationary
(in any sense you like) and has mean zero.

Any comments or suggestions about how deal with this kind of question
would be greatly appreciated.

Thanks in advance. cgm

(Apologies if this is mildly offtopic.)

I'm looking for examples of uses of the symbol which looks like a
superimposed \prod and \coprod (or, more accurately, a product
operator whose bottom half is a mirror copy of the top half by a
horizontal axis). I'd like to suggest the addition of this character
to Unicode/ISO-10646, so the more examples I can find the more
convincing the proposal will be.

See http://www.madore.org/~david/.misc/biproduct/test.pdf > (or
test.png in the same directory) for an example (made up) formula using
this symbol. The files biproduct.{pfb,tfm} in the same directory
contain a font having the character in question.

Hi all,

I am writing to seek some insights to prove the following Conjecture:

Given integers n > m > k >1 , let p(x) be the n-th degree polynomial
p(x) = x^(n-m) * ( x-a_1)*(x-a_2)*..(x-a_m),
where a_i are strictly positive real numbers. Let f(x) be the (n-k)-
th
derivative of p(x), and let b_1, b_2,..., b_k be the zeros of f(x),
ordered by
b_1 >= b_2 ... >= b_k. Then, the choice of {a_1, a_2, ..., a_m}
that
minimizes the ratio (b_1 / b_k) is when the a's are all equal, ie,
a_1 = a_2=...=a_m.

Of course, Gauss-Lucas theorem tels us that the b's are all real and
lie between 0 and max{a_1,...,a_m}. I am interested in the ratio of
the maximum root (b_1) and the minimum root (b_k) of f(x).

I have investigated using several results from "Analytical theory of
polynomials" be Rahman and Schmeisser, but so far has had no success.
If you could provide some insights / reference to similar results,
that would be greatly appreciated!!

Hi. Let F be a subfield of the complex numbers C, equipped with a
field automorphism h:F-->F satisfying hh=identity. (So h is either
order 2 or order 1.) Assume that h satisfies a nondegeneracy property:
for all nonzero finite sequences a, b, ..., c of elements of F,
a.h(a) + b.h(b) + ... + c.h(c) =/= 0.
Does there always exist a field embedding e:F-->C with respect to
which h is identified with complex conjugation, so that
e(h(x))=(e(x))* for all x in F? Or, equivalently: is there a field
embedding

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