1,617 research outputs found
Local-global principles for Galois cohomology
This paper proves local-global principles for Galois cohomology groups over
function fields of curves that are defined over a complete discretely
valued field. We show in particular that such principles hold for , for all . This is motivated by work of Kato and others, where
such principles were shown in related cases for . Using our results in
combination with cohomological invariants, we obtain local-global principles
for torsors and related algebraic structures over . Our arguments rely on
ideas from patching as well as the Bloch-Kato conjecture.Comment: 32 pages. Some changes of notation. Statement of Lemma 2.4.4
corrected. Lemma 3.3.2 strengthened and made a proposition. Some proofs
modified to fix or clarify specific points or to streamline the presentatio
Plancherel formula for and applications to the Ichino-Ikeda and formal degree conjectures for unitary groups
We establish an explicit Plancherel decomposition for
where is a quadratic
extension of local fields of characteristic zero by making use of a local
functional equation for Asai -factors. We also give two applications of
this Plancherel formula: first to the global Ichino-Ikeda conjecture for
unitary groups by completing a comparison between local relative characters
that was left open by W. Zhang and secondly to the Hiraga-Ichino-Ikeda
conjecture on formal degrees in the case of unitary groups
The feasible region of induced graphs
The feasible region of a graph is the collection
of points in the unit square such that there exists a sequence of
graphs whose edge densities approach and whose induced -densities
approach . A complete description of is not known
for any with at least four vertices that is not a clique or an independent
set. The feasible region provides a lot of combinatorial information about .
For example, the supremum of over all is
the inducibility of and yields the Kruskal-Katona
and clique density theorems.
We begin a systematic study of by proving some
general statements about the shape of and giving
results for some specific graphs . Many of our theorems apply to the more
general setting of quantum graphs. For example, we prove a bound for quantum
graphs that generalizes an old result of Bollob\'as for the number of cliques
in a graph with given edge density. We also consider the problems of
determining when , is a star, or is a
complete bipartite graph. In the case of our results sharpen those
predicted by the edge-statistics conjecture of Alon et. al. while also
extending a theorem of Hirst for that was proved using computer aided
techniques and flag algebras. The case of the 4-cycle seems particularly
interesting and we conjecture that is determined by
the solution to the triangle density problem, which has been solved by
Razborov.Comment: 27 page
C-polynomials and LC-functions: towards a generalization of the Hurwitz zeta function
Let be an analytic function
at , let be the
sequence of Appell polynomials, which we call , constructed from the sequence of the coefficients of , and
let be the sequence of C-polynomials associated to the function
which we call . This work addresses three main topics. The first concerns the study of
these two types of polynomials and the connection between them. In the second,
inspired by the definition of the P-polynomials and under an additional
condition on , we introduce and study a function of complex
variables which generalizes the function and which we denote by
. In the third part we generalize the Hurwitz zeta function as well
as its fundamental properties, the most remarkable being the Hurwitz's formula,
by constructing a new class of functions related to the C-polynomials and which we call
( being a positive integer
depending on the choice of ).Comment: 33 pages, 6 figure
Adaptive procedures in convolution models with known or partially known noise distribution
In a convolution model, we observe random variables whose distribution is the
convolution of some unknown density f and some known or partially known noise
density g. In this paper, we focus on statistical procedures, which are
adaptive with respect to the smoothness parameter tau of unknown density f, and
also (in some cases) to some unknown parameter of the noise density g. In a
first part, we assume that g is known and polynomially smooth. We provide
goodness-of-fit procedures for the test H_0:f=f_0, where the alternative H_1 is
expressed with respect to L_2-norm. Our adaptive (w.r.t tau) procedure behaves
differently according to whether f_0 is polynomially or exponentially smooth. A
payment for adaptation is noted in both cases and for computing this, we
provide a non-uniform Berry-Esseen type theorem for degenerate U-statistics. In
the first case we prove that the payment for adaptation is optimal (thus
unavoidable). In a second part, we study a wider framework: a semiparametric
model, where g is exponentially smooth and stable, and its self-similarity
index s is unknown. In order to ensure identifiability, we restrict our
attention to polynomially smooth, Sobolev-type densities f. In this context, we
provide a consistent estimation procedure for s. This estimator is then
plugged-into three different procedures: estimation of the unknown density f,
of the functional \int f^2 and test of the hypothesis H_0. These procedures are
adaptive with respect to both s and tau and attain the rates which are known
optimal for known values of s and tau. As a by-product, when the noise is known
and exponentially smooth our testing procedure is adaptive for testing
Sobolev-type densities.Comment: 35 pages + annexe de 8 page
A convenient implementation of the overlap between arbitrary Hartree-Fock-Bogoliubov vacua for projection
Overlap between Hartree-Fock-Bogoliubov(HFB) vacua is very important in the
beyond mean-field calculations. However, in the HFB transformation, the
matrices are sometimes singular due to the exact emptiness () or full
occupation () of some single-particle orbits. This singularity may cause
some problem in evaluating the overlap between HFB vacua through Pfaffian. We
found that this problem can be well avoided by setting those zero occupation
numbers to some tiny values (e.g., ). This treatment does not
change the HFB vacuum state because are numerically zero
relative to 1. Therefore, for arbitrary HFB transformation, we say that the
matrices can always be nonsingular. From this standpoint, we present a
new convenient Pfaffian formula for the overlap between arbitrary HFB vacua,
which is especially suitable for symmetry restoration. Testing calculations
have been performed for this new formula. It turns out that our method is
reliable and accurate in evaluating the overlap between arbitrary HFB vacua.Comment: 5 pages, 2 figures. Published versio
- …