Zero modes, gauge fixing, monodromies, ζ\zeta-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}$=-d^2/d\tau^2+\ddot g/g$, $\ddot g\equiv d^2g/d\tau^2$, with a generic function $g(\tau)$, 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

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

Differentially Private Distributed Optimization

In distributed optimization and iterative consensus literature, a standard problem is for $N$ agents to minimize a function $f$ over a subset of Euclidean space, where the cost function is expressed as a sum $\sum f_i$. 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 $\epsilon$-differential privacy the accuracy of the algorithm has the order of $O(\frac{1}{\epsilon^2})$

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

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

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

Quantum Effective Action in Spacetimes with Branes and Boundaries

We construct quantum effective action in spacetime with branes/boundaries. This construction is based on the reduction of the underlying Neumann type boundary value problem for the propagator of the theory to that of the much more manageable Dirichlet problem. In its turn, this reduction follows from the recently suggested Neumann-Dirichlet duality which we extend beyond the tree level approximation. In the one-loop approximation this duality suggests that the functional determinant of the differential operator subject to Neumann boundary conditions in the bulk factorizes into the product of its Dirichlet counterpart and the functional determinant of a special operator on the brane -- the inverse of the brane-to-brane propagator. As a byproduct of this relation we suggest a new method for surface terms of the heat kernel expansion. This method allows one to circumvent well-known difficulties in heat kernel theory on manifolds with boundaries for a wide class of generalized Neumann boundary conditions. In particular, we easily recover several lowest order surface terms in the case of Robin and oblique boundary conditions. We briefly discuss multi-loop applications of the suggested Dirichlet reduction and the prospects of constructing the universal background field method for systems with branes/boundaries, analogous to the Schwinger-DeWitt technique.Comment: LaTeX, 25 pages, final version, to appear in Phys. Rev.

Implementation of an Optimal First-Order Method for Strongly Convex Total Variation Regularization

We present a practical implementation of an optimal first-order method, due to Nesterov, for large-scale total variation regularization in tomographic reconstruction, image deblurring, etc. The algorithm applies to $\mu$-strongly convex objective functions with $L$-Lipschitz continuous gradient. In the framework of Nesterov both $\mu$ and $L$ are assumed known -- an assumption that is seldom satisfied in practice. We propose to incorporate mechanisms to estimate locally sufficient $\mu$ and $L$ during the iterations. The mechanisms also allow for the application to non-strongly convex functions. We discuss the iteration complexity of several first-order methods, including the proposed algorithm, and we use a 3D tomography problem to compare the performance of these methods. The results show that for ill-conditioned problems solved to high accuracy, the proposed method significantly outperforms state-of-the-art first-order methods, as also suggested by theoretical results.Comment: 23 pages, 4 figure