2,495 research outputs found
On the resonances of the Laplacian on waveguides
The resonances for the Dirichlet and Neumann Laplacian are studied on
compactly perturbed waveguides. An upper bound on the number of resonances near
the physical plane is proven. In the absence of resonances, an upper bound is
proven for the localised resolvent. This is then used to prove that the
existence of a quasimode whose asymptotics is bounded away from the thresholds
implies the existence of resonances converging to the real axis
Stability and integration over Bergman metrics
We study partition functions of random Bergman metrics, with the actions
defined by a class of geometric functionals known as `stability functions'. We
introduce a new stability invariant - the critical value of the coupling
constant - defined as the minimal coupling constant for which the partition
function converges. It measures the minimal degree of stability of geodesic
rays in the space the Bergman metrics, with respect to the action. We calculate
this critical value when the action is the -balancing energy, and show
that on a Riemann surface of genus .Comment: 24 pages, 3 figure
Weil-Petersson perspectives
We highlight recent progresses in the study of the Weil-Petersson (WP)
geometry of finite dimensional Teichm\"{u}ller spaces. For recent progress on
and the understanding of infinite dimensional Teichm\"{u}ller spaces the reader
is directed to the recent work of Teo-Takhtajan. As part of the highlight, we
also present possible directions for future investigations.Comment: 18 page
Robust Inference Under Heteroskedasticity via the Hadamard Estimator
Drawing statistical inferences from large datasets in a model-robust way is
an important problem in statistics and data science. In this paper, we propose
methods that are robust to large and unequal noise in different observational
units (i.e., heteroskedasticity) for statistical inference in linear
regression. We leverage the Hadamard estimator, which is unbiased for the
variances of ordinary least-squares regression. This is in contrast to the
popular White's sandwich estimator, which can be substantially biased in high
dimensions. We propose to estimate the signal strength, noise level,
signal-to-noise ratio, and mean squared error via the Hadamard estimator. We
develop a new degrees of freedom adjustment that gives more accurate confidence
intervals than variants of White's sandwich estimator. Moreover, we provide
conditions ensuring the estimator is well-defined, by studying a new random
matrix ensemble in which the entries of a random orthogonal projection matrix
are squared. We also show approximate normality, using the second-order
Poincare inequality. Our work provides improved statistical theory and methods
for linear regression in high dimensions
Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive
We investigate the set a) of positive, trace preserving maps acting on
density matrices of size N, and a sequence of its nested subsets: the sets of
maps which are b) decomposable, c) completely positive, d) extended by identity
impose positive partial transpose and e) are superpositive. Working with the
Hilbert-Schmidt (Euclidean) measure we derive tight explicit two-sided bounds
for the volumes of all five sets. A sample consequence is the fact that, as N
increases, a generic positive map becomes not decomposable and, a fortiori, not
completely positive.
Due to the Jamiolkowski isomorphism, the results obtained for quantum maps
are closely connected to similar relations between the volume of the set of
quantum states and the volumes of its subsets (such as states with positive
partial transpose or separable states) or supersets. Our approach depends on
systematic use of duality to derive quantitative estimates, and on various
tools of classical convexity, high-dimensional probability and geometry of
Banach spaces, some of which are not standard.Comment: 34 pages in Latex including 3 figures in eps, ver 2: minor revision
Regulators of rank one quadratic twists
We investigate the regulators of elliptic curves with rank 1 in some families
of quadratic twists of a fixed elliptic curve. In particular, we formulate some
conjectures on the average size of these regulators. We also describe an
efficient algorithm to compute explicitly some of the invariants of an odd
quadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevich
group, etc.) and we discuss the numerical data that we obtain and compare it
with our predictions.Comment: 28 pages with 32 figure
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