140,961 research outputs found
Partition Functions in Even Dimensional AdS via Quasinormal Mode Methods
In this note, we calculate the one-loop determinant for a massive scalar
(with conformal dimension ) in even-dimensional AdS space,
using the quasinormal mode method developed in arXiv:0908.2657 by Denef,
Hartnoll, and Sachdev. Working first in two dimensions on the related Euclidean
hyperbolic plane , we find a series of zero modes for negative real values
of whose presence indicates a series of poles in the one-loop
partition function in the complex plane; these poles
contribute temperature-independent terms to the thermal AdS partition function
computed in arXiv:0908.2657. Our results match those in a series of papers by
Camporesi and Higuchi, as well as Gopakumar et.al. in arXiv:1103.3627 and
Banerjee et.al. in arXiv:1005.3044. We additionally examine the meaning of
these zero modes, finding that they Wick-rotate to quasinormal modes of the
AdS black hole. They are also interpretable as matrix elements of the
discrete series representations of in the space of smooth functions
on . We generalize our results to general even dimensional AdS,
again finding a series of zero modes which are related to discrete series
representations of , the motion group of .Comment: 27 pages; v2: minor updates and JHEP versio
Robust and efficient solution of the drum problem via Nystrom approximation of the Fredholm determinant
The drum problem-finding the eigenvalues and eigenfunctions of the Laplacian
with Dirichlet boundary condition-has many applications, yet remains
challenging for general domains when high accuracy or high frequency is needed.
Boundary integral equations are appealing for large-scale problems, yet certain
difficulties have limited their use. We introduce two ideas to remedy this: 1)
We solve the resulting nonlinear eigenvalue problem using Boyd's method for
analytic root-finding applied to the Fredholm determinant. We show that this is
many times faster than the usual iterative minimization of a singular value. 2)
We fix the problem of spurious exterior resonances via a combined field
representation. This also provides the first robust boundary integral
eigenvalue method for non-simply-connected domains. We implement the new method
in two dimensions using spectrally accurate Nystrom product quadrature. We
prove exponential convergence of the determinant at roots for domains with
analytic boundary. We demonstrate 13-digit accuracy, and improved efficiency,
in a variety of domain shapes including ones with strong exterior resonances.Comment: 21 pages, 7 figures, submitted to SIAM Journal of Numerical Analysis.
Updated a duplicated picture. All results unchange
Asymptotics via Steepest Descent for an Operator Riemann-Hilbert Problem
In this paper, we take the first step towards an extension of the nonlinear
steepest descent method of Deift, Its and Zhou to the case of operator
Riemann-Hilbert problems. In particular, we provide long range asymptotics for
a Fredholm determinant arising in the computation of the probability of finding
a string of n adjacent parallel spins up in the antiferromagnetic ground state
of the spin 1/2 XXX Heisenberg Chain. Such a determinant can be expressed in
terms of the solution of an operator Riemann-Hilbert factorization problem
Resultants and Moving Surfaces
We prove a conjectured relationship among resultants and the determinants
arising in the formulation of the method of moving surfaces for computing the
implicit equation of rational surfaces formulated by Sederberg. In addition, we
extend the validity of this method to the case of not properly parametrized
surfaces without base points.Comment: 21 pages, LaTex, uses academic.cls. To appear: Journal of Symbolic
Computatio
Nearly Optimal Private Convolution
We study computing the convolution of a private input with a public input
, while satisfying the guarantees of -differential
privacy. Convolution is a fundamental operation, intimately related to Fourier
Transforms. In our setting, the private input may represent a time series of
sensitive events or a histogram of a database of confidential personal
information. Convolution then captures important primitives including linear
filtering, which is an essential tool in time series analysis, and aggregation
queries on projections of the data.
We give a nearly optimal algorithm for computing convolutions while
satisfying -differential privacy. Surprisingly, we follow
the simple strategy of adding independent Laplacian noise to each Fourier
coefficient and bounding the privacy loss using the composition theorem of
Dwork, Rothblum, and Vadhan. We derive a closed form expression for the optimal
noise to add to each Fourier coefficient using convex programming duality. Our
algorithm is very efficient -- it is essentially no more computationally
expensive than a Fast Fourier Transform.
To prove near optimality, we use the recent discrepancy lowerbounds of
Muthukrishnan and Nikolov and derive a spectral lower bound using a
characterization of discrepancy in terms of determinants
Implicitization of rational surfaces using toric varieties
A parameterized surface can be represented as a projection from a certain
toric surface. This generalizes the classical homogeneous and bihomogeneous
parameterizations. We extend to the toric case two methods for computing the
implicit equation of such a rational parameterized surface. The first approach
uses resultant matrices and gives an exact determinantal formula for the
implicit equation if the parameterization has no base points. In the case the
base points are isolated local complete intersections, we show that the
implicit equation can still be recovered by computing any non-zero maximal
minor of this matrix.
The second method is the toric extension of the method of moving surfaces,
and involves finding linear and quadratic relations (syzygies) among the input
polynomials. When there are no base points, we show that these can be put
together into a square matrix whose determinant is the implicit equation. Its
extension to the case where there are base points is also explored.Comment: 28 pages, 1 figure. Numerous major revisions. New proof of method of
moving surfaces. Paper accepted and to appear in Journal of Algebr
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