2,328 research outputs found
Zero modes, gauge fixing, monodromies, -functions and all that
We discuss various issues associated with the calculation of the reduced
functional determinant of a special second order differential operator
\boldmath{F}, , with a
generic function , subject to periodic and Dirichlet boundary
conditions. These issues include the gauge-fixed path integral representation
of this determinant, the monodromy method of its calculation and the
combination of the heat kernel and zeta-function technique for the derivation
of its period dependence. Motivations for this particular problem, coming from
applications in quantum cosmology, are also briefly discussed. They include the
problem of microcanonical initial conditions in cosmology driven by a conformal
field theory, cosmological constant and cosmic microwave background problems.Comment: 17 pages, to appear in J. Phys. A: Math. Theor. arXiv admin note:
substantial text overlap with arXiv:1111.447
Differentially Private Distributed Optimization
In distributed optimization and iterative consensus literature, a standard
problem is for agents to minimize a function over a subset of Euclidean
space, where the cost function is expressed as a sum . In this paper,
we study the private distributed optimization (PDOP) problem with the
additional requirement that the cost function of the individual agents should
remain differentially private. The adversary attempts to infer information
about the private cost functions from the messages that the agents exchange.
Achieving differential privacy requires that any change of an individual's cost
function only results in unsubstantial changes in the statistics of the
messages. We propose a class of iterative algorithms for solving PDOP, which
achieves differential privacy and convergence to the optimal value. Our
analysis reveals the dependence of the achieved accuracy and the privacy levels
on the the parameters of the algorithm. We observe that to achieve
-differential privacy the accuracy of the algorithm has the order of
Multi-step Fermi normal coordinates
We generalize the concept of Fermi normal coordinates adapted to a geodesic
to the case where the tangent space to the manifold at the base point is
decomposed into a direct product of an arbitrary number of subspaces, so that
we follow several geodesics in turn to find the point with given coordinates.
We compute the connection and the metric as integrals of the Riemann tensor. In
the case of one subspace (Riemann normal coordinates) or two subspaces, we
recover some results previously found by Nesterov, using somewhat different
techniques.Comment: 9 pages, 4 figure
Quasigroups, Asymptotic Symmetries and Conservation Laws in General Relativity
A new quasigroup approach to conservation laws in general relativity is
applied to study asymptotically flat at future null infinity spacetime. The
infinite-parametric Newman-Unti group of asymptotic symmetries is reduced to
the Poincar\'e quasigroup and the Noether charge associated with any element of
the Poincar\'e quasialgebra is defined. The integral conserved quantities of
energy-momentum and angular momentum are linear on generators of Poincar\'e
quasigroup, free of the supertranslation ambiguity, posess the flux and
identically equal to zero in Minkowski spacetime.Comment: RevTeX4, 5 page
Solving rank-constrained semidefinite programs in exact arithmetic
We consider the problem of minimizing a linear function over an affine
section of the cone of positive semidefinite matrices, with the additional
constraint that the feasible matrix has prescribed rank. When the rank
constraint is active, this is a non-convex optimization problem, otherwise it
is a semidefinite program. Both find numerous applications especially in
systems control theory and combinatorial optimization, but even in more general
contexts such as polynomial optimization or real algebra. While numerical
algorithms exist for solving this problem, such as interior-point or
Newton-like algorithms, in this paper we propose an approach based on symbolic
computation. We design an exact algorithm for solving rank-constrained
semidefinite programs, whose complexity is essentially quadratic on natural
degree bounds associated to the given optimization problem: for subfamilies of
the problem where the size of the feasible matrix is fixed, the complexity is
polynomial in the number of variables. The algorithm works under assumptions on
the input data: we prove that these assumptions are generically satisfied. We
also implement it in Maple and discuss practical experiments.Comment: Published at ISSAC 2016. Extended version submitted to the Journal of
Symbolic Computatio
Nonlinear interfaces: intrinsically nonparaxial regimes and effects
The behaviour of optical solitons at planar nonlinear boundaries is a problem rich in intrinsically nonparaxial regimes that cannot be fully addressed by theories based on the nonlinear Schrödinger equation. For instance, large propagation angles are typically involved in external refraction at interfaces. Using a recently proposed generalized Snell's law for Helmholtz solitons, we analyse two such effects: nonlinear external refraction and total internal reflection at interfaces where internal and external refraction, respectively, would be found in the absence of nonlinearity. The solutions obtained from the full numerical integration of the nonlinear Helmholtz equation show excellent agreement with the theoretical predictions
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