79,077 research outputs found
Additive twists of Fourier coefficients of symmetric-square lifts
We study the sum of additively twisted Fourier coefficients of a
symmetric-square lift of a Maass form invariant under the full modular group.
Our bounds are uniform in terms of the spectral parameter of the Maass form, as
well as in terms of the additive twist.Comment: 13 pages. v2: fixed the relation between T and t_j on p.2 and added
clarification to some reference
A Direct Limit for Limit Hilbert-Kunz Multiplicity for Smooth Projective Curves
This paper concerns the question of whether a more direct limit can be used
to obtain the limit Hilbert-Kunz multiplicity, a possible candidate for a
characteristic zero Hilbert-Kunz multiplicity. The main goal is to establish an
affirmative answer for one of the main cases for which the limit Hilbert-Kunz
multiplicity is even known to exist, namely that of graded ideals in the
homogeneous coordinate ring of smooth projective curves. The proof involves
more careful estimates of bounds found independently by Brenner and Trivedi on
the dimensions of the cohomologies of twists of the syzygy bundle as the
characteristic p goes to infinity and uses asymptotic results of Trivedi on the
slopes of Harder-Narasimham filtrations of Frobenius pullbacks of bundles. In
view of unpublished results of Gessel and Monsky, the case of maximal ideals in
diagonal hypersurfaces is also discussed in depth.Comment: Minor typos fixe
On the (non)rigidity of the Frobenius Endomorphism over Gorenstein Rings
It is well-known that for a large class of local rings of positive
characteristic, including complete intersection rings, the Frobenius
endomorphism can be used as a test for finite projective dimension. In this
paper, we exploit this property to study the structure of such rings. One of
our results states that the Picard group of the punctured spectrum of such a
ring cannot have -torsion. When is a local complete intersection,
this recovers (with a purely local algebra proof) an analogous statement for
complete intersections in projective spaces first given in SGA and also a
special case of a conjecture by Gabber. Our method also leads to many simply
constructed examples where rigidity for the Frobenius endomorphism does not
hold, even when the rings are Gorenstein with isolated singularity. This is in
stark contrast to the situation for complete intersection rings. Also, a
related length criterion for modules of finite length and finite projective
dimension is discussed towards the end.Comment: Minor changes in Example 2.2 and Theorem 2.9. Conjecture 1.2 was
added
Iterative Row Sampling
There has been significant interest and progress recently in algorithms that
solve regression problems involving tall and thin matrices in input sparsity
time. These algorithms find shorter equivalent of a n*d matrix where n >> d,
which allows one to solve a poly(d) sized problem instead. In practice, the
best performances are often obtained by invoking these routines in an iterative
fashion. We show these iterative methods can be adapted to give theoretical
guarantees comparable and better than the current state of the art.
Our approaches are based on computing the importances of the rows, known as
leverage scores, in an iterative manner. We show that alternating between
computing a short matrix estimate and finding more accurate approximate
leverage scores leads to a series of geometrically smaller instances. This
gives an algorithm that runs in
time for any , where the term is comparable
to the cost of solving a regression problem on the small approximation. Our
results are built upon the close connection between randomized matrix
algorithms, iterative methods, and graph sparsification.Comment: 26 pages, 2 figure
When bigger isn’t better : bailouts and bank behaviour
Lending retail deposits to SMEs and household borrowers may be the traditional role of
commercial banks: but banking in Britain has been transformed by increasing consolidation
and by the lure of high returns available from wholesale Investment activities. With
appropriate changes to the baseline model of commercial banking in Allen and Gale (2007),
we show how market power enables banks to collect „seigniorage‟; and how „tail risk‟
investment allows losses to be shifted onto the taxpayer.
In principle, the high franchise values associated with market power assist regulatory capital
requirements to check risk-taking. But when big banks act strategically, bailout expectations
can undermine these disciplining devices: and the taxpayer ends up „on the hook‟- as in the
recent crisis. That structural change is needed to prevent a repeat seems clear from the
Vickers report, which proposes to protect the taxpayer by a „ring fence‟separating
commercial and investment banking
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