222,805 research outputs found
Convex Q Learning in a Stochastic Environment: Extended Version
The paper introduces the first formulation of convex Q-learning for Markov
decision processes with function approximation. The algorithms and theory rest
on a relaxation of a dual of Manne's celebrated linear programming
characterization of optimal control. The main contributions firstly concern
properties of the relaxation, described as a deterministic convex program: we
identify conditions for a bounded solution, and a significant relationship
between the solution to the new convex program, and the solution to standard
Q-learning. The second set of contributions concern algorithm design and
analysis: (i) A direct model-free method for approximating the convex program
for Q-learning shares properties with its ideal. In particular, a bounded
solution is ensured subject to a simple property of the basis functions; (ii)
The proposed algorithms are convergent and new techniques are introduced to
obtain the rate of convergence in a mean-square sense; (iii) The approach can
be generalized to a range of performance criteria, and it is found that
variance can be reduced by considering ``relative'' dynamic programming
equations; (iv) The theory is illustrated with an application to a classical
inventory control problem.Comment: Extended version of "Convex Q-learning in a stochastic environment",
IEEE Conference on Decision and Control, 2023 (to appear
S-Duality and Exact Type IIB Superstring Backgrounds
A geometrical approach in the non-symmetric connection framework is employed
to examine the issue of higher order corrections to D=10 type IIB
superstring backgrounds with a covariantly constant null Killing isometry and
non-zero Ramond-Ramond field content. These describe generalized supersymmetric
string waves and were obtained recently by us through the S-duality
transformation of purely NS-NS plane wave backgrounds. We find that the
backgrounds are exact subject to the existence of certain field redefinitions
and provided certain restrictive conditions are satisfied.Comment: 21 Pages, LATEX fil
Strong Coupling Theory for Interacting Lattice Models
We develop a strong coupling approach for a general lattice problem. We argue
that this strong coupling perspective represents the natural framework for a
generalization of the dynamical mean field theory (DMFT). The main result of
this analysis is twofold: 1) It provides the tools for a unified treatment of
any non-local contribution to the Hamiltonian. Within our scheme, non-local
terms such as hopping terms, spin-spin interactions, or non-local Coulomb
interactions are treated on equal footing. 2) By performing a detailed
strong-coupling analysis of a generalized lattice problem, we establish the
basis for possible clean and systematic extensions beyond DMFT. To this end, we
study the problem using three different perspectives. First, we develop a
generalized expansion around the atomic limit in terms of the coupling
constants for the non-local contributions to the Hamiltonian. By analyzing the
diagrammatics associated with this expansion, we establish the equations for a
generalized dynamical mean-field theory (G-DMFT). Second, we formulate the
theory in terms of a generalized strong coupling version of the Baym-Kadanoff
functional. Third, following Pairault, Senechal, and Tremblay, we present our
scheme in the language of a perturbation theory for canonical fermionic and
bosonic fields and we establish the interpretation of various strong coupling
quantities within a standard perturbative picture.Comment: Revised Version, 17 pages, 5 figure
Unified Einstein-Virasoro Master Equation in the General Non-Linear Sigma Model
The Virasoro master equation (VME) describes the general affine-Virasoro
construction T=L^{ab}J_aJ_b+iD^a \dif J_a in the operator algebra of the WZW
model, where is the inverse inertia tensor and is the
improvement vector. In this paper, we generalize this construction to find the
general (one-loop) Virasoro construction in the operator algebra of the general
non-linear sigma model. The result is a unified Einstein-Virasoro master
equation which couples the spacetime spin-two field to the background
fields of the sigma model. For a particular solution , the unified
system reduces to the canonical stress tensors and conventional Einstein
equations of the sigma model, and the system reduces to the general
affine-Virasoro construction and the VME when the sigma model is taken to be
the WZW action. More generally, the unified system describes a space of
conformal field theories which is presumably much larger than the sum of the
general affine-Virasoro construction and the sigma model with its canonical
stress tensors. We also discuss a number of algebraic and geometrical
properties of the system, including its relation to an unsolved problem in the
theory of -structures on manifolds with torsion.Comment: LaTeX, 55 pages, one postscript figure, uses epsfig.sty. contains a
few minor corrections; version to be published in Int. J. Mod. Phys.
Generalized conformal structure, dilaton gravity and SYK
A theory admits generalized conformal structure if the only scale in the
quantum theory is set by a dimensionful coupling. SYK is an example of a theory
with generalized conformal structure and in this paper we investigate the
consequences of this structure for correlation functions and for the
holographic realization of SYK. The Ward identities associated with the
generalized conformal structure of SYK are implemented holographically in
gravity/multiple scalar theories, which always have a parent AdS origin.
For questions involving only the graviton/running scalar sector, one can always
describe the bulk running in terms of a single scalar but multiple running
scalars are in general needed once one includes the bulk fields corresponding
to all SYK operators. We then explore chaos in holographic theories with
generalized conformal structure. The four point function explored by Maldacena,
Shenker and Stanford exhibits exactly the same chaotic behaviour in any such
theory as in holographic realizations of conformal theories i.e. the
dimensionful coupling scale does not affect the chaotic exponential growth.Comment: 42 pages, 3 figure
Finite Temperature Models of Bose-Einstein Condensation
The theoretical description of trapped weakly-interacting Bose-Einstein
condensates is characterized by a large number of seemingly very different
approaches which have been developed over the course of time by researchers
with very distinct backgrounds. Newcomers to this field, experimentalists and
young researchers all face a considerable challenge in navigating through the
`maze' of abundant theoretical models, and simple correspondences between
existing approaches are not always very transparent. This Tutorial provides a
generic introduction to such theories, in an attempt to single out common
features and deficiencies of certain `classes of approaches' identified by
their physical content, rather than their particular mathematical
implementation.
This Tutorial is structured in a manner accessible to a non-specialist with a
good working knowledge of quantum mechanics. Although some familiarity with
concepts of quantum field theory would be an advantage, key notions such as the
occupation number representation of second quantization are nonetheless briefly
reviewed. Following a general introduction, the complexity of models is
gradually built up, starting from the basic zero-temperature formalism of the
Gross-Pitaevskii equation. This structure enables readers to probe different
levels of theoretical developments (mean-field, number-conserving and
stochastic) according to their particular needs. In addition to its `training
element', we hope that this Tutorial will prove useful to active researchers in
this field, both in terms of the correspondences made between different
theoretical models, and as a source of reference for existing and developing
finite-temperature theoretical models.Comment: Detailed Review Article on finite temperature theoretical techniques
for studying weakly-interacting atomic Bose-Einstein condensates written at
an elementary level suitable for non-experts in this area (e.g. starting PhD
students). Now includes table of content
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