3,787 research outputs found
The Kadison-Singer Problem in Mathematics and Engineering
We will show that the famous, intractible 1959 Kadison-Singer problem in
-algebras is equivalent to fundamental unsolved problems in a dozen
areas of research in pure mathematics, applied mathematics and Engineering.
This gives all these areas common ground on which to interact as well as
explaining why each of these areas has volumes of literature on their
respective problems without a satisfactory resolution. In each of these areas
we will reduce the problem to the minimum which needs to be proved to solve
their version of Kadison-Singer. In some areas we will prove what we believe
will be the strongest results ever available in the case that Kadison-Singer
fails. Finally, we will give some directions for constructing a counter-example
to Kadison-Singer
Stark points and Hida-Rankin p-adic L-function
This article is devoted to the elliptic Stark conjecture formulated by
Darmon, Lauder and Rotger [DLR], which proposes a formula for the
transcendental part of a -adic avatar of the leading term at of the
Hasse-Weil-Artin -series of an elliptic
curve twisted by the tensor product of two odd
-dimensional Artin representations, when the order of vanishing is two. The
main ingredient of this formula is a -adic regulator involving
the -adic formal group logarithm of suitable Stark points on . This
conjecture was proved in [DLR] in the setting where and
are induced from characters of the same imaginary quadratic field . In this
note we prove a refinement of this result, that was discovered experimentally
in Remark 3.4 of [DLR] in a few examples. Namely, we are able to determine the
algebraic constant up to which the main theorem of [DLR] holds in a particular
setting where the Hida-Rankin -adic -function associated to a pair of
Hida families can be exploited to provide an alternative proof of the same
result. This constant encodes local and global invariants of both and
On the undefinability of Tsirelson's space and its descendants
We prove that Tsirelson's space cannot be defined explicitly from the
classical Banach sequence spaces. We also prove that any Banach space that is
explicitly definable from a class of spaces that contain or must
contain or as well
A characterization of Schauder frames which are near-Schauder bases
A basic problem of interest in connection with the study of Schauder frames
in Banach spaces is that of characterizing those Schauder frames which can
essentially be regarded as Schauder bases. In this paper, we give a solution to
this problem using the notion of the minimal-associated sequence spaces and the
minimal-associated reconstruction operators for Schauder frames. We prove that
a Schauder frame is a near-Schauder basis if and only if the kernel of the
minimal-associated reconstruction operator contains no copy of . In
particular, a Schauder frame of a Banach space with no copy of is a
near-Schauder basis if and only if the minimal-associated sequence space
contains no copy of . In these cases, the minimal-associated
reconstruction operator has a finite dimensional kernel and the dimension of
the kernel is exactly the excess of the near-Schauder basis. Using these
results, we make related applications on Besselian frames and near-Riesz bases.Comment: 12 page
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