Basic Concepts Underlying Singular Perturbation Techniques

In many singular perturbation problems multiple scales are used. For instance, one may use both the coordinate x and the coordinate x^* = ε^(-1)x. In a secular-type problem x and x^* are used simultaneously. This paper discusses layer-type problems in which x^* is used in a thin layer and x outside this layer. Assume one seeks approximations to a function f(x,ε), uniformly valid to some order in ε for x in a closed interval D. In layer-type problems one uses (at least) two expansions (called inner and outer) neither of which is uniformly valid but whose domains of validity together cover the interval D. To define "domain of validity" one needs to consider intervals whose endpoints depend on epsilon. In the construction of the inner and outer expansions, constants and functions of e occur which are determined by comparison of the two expansions "matching." The comparison is possible only in the domain of overlap of their regions of validity. Once overlap is established, matching is easily carried out. Heuristic ideas for determining domains of validity of approximations by a study of the corresponding equations are illustrated with the aid of model equations. It is shown that formally small terms in an equation may have large integrated effects. The study of this is of central importance for understanding layer-type problems. It is emphasized that considering the expansions as the result of applying limit processes can lead to serious errors and, in any case, hides the nature of the expansions

Adversarial Sampling and Training for Semi-Supervised Information Retrieval

Ad-hoc retrieval models with implicit feedback often have problems, e.g., the imbalanced classes in the data set. Too few clicked documents may hurt generalization ability of the models, whereas too many non-clicked documents may harm effectiveness of the models and efficiency of training. In addition, recent neural network-based models are vulnerable to adversarial examples due to the linear nature in them. To solve the problems at the same time, we propose an adversarial sampling and training framework to learn ad-hoc retrieval models with implicit feedback. Our key idea is (i) to augment clicked examples by adversarial training for better generalization and (ii) to obtain very informational non-clicked examples by adversarial sampling and training. Experiments are performed on benchmark data sets for common ad-hoc retrieval tasks such as Web search, item recommendation, and question answering. Experimental results indicate that the proposed approaches significantly outperform strong baselines especially for high-ranked documents, and they outperform IRGAN in NDCG@5 using only 5% of labeled data for the Web search task.Comment: Published in WWW 201

Resumming the string perturbation series

We use the AdS/CFT correspondence to study the resummation of a perturbative genus expansion appearing in the type II superstring dual of ABJM theory. Although the series is Borel summable, its Borel resummation does not agree with the exact non-perturbative answer due to the presence of complex instantons. The same type of behavior appears in the WKB quantization of the quartic oscillator in Quantum Mechanics, which we analyze in detail as a toy model for the string perturbation series. We conclude that, in these examples, Borel summability is not enough for extracting non-perturbative information, due to non-perturbative effects associated to complex instantons. We also analyze the resummation of the genus expansion for topological string theory on local $\mathbb P^1 \times \mathbb P^1$, which is closely related to ABJM theory. In this case, the non-perturbative answer involves membrane instantons computed by the refined topological string, which are crucial to produce a well-defined result. We give evidence that the Borel resummation of the perturbative series requires such a non-perturbative sector.Comment: 31 pages, 9 figures; v3 : clarifications added and misprints correcte

Lessons in quantum gravity from quantum field theory

This paper reviews advances in the understanding of quantum gravity based on field theory calculations in the AdS/CFT correspondence.Comment: 18 pages, Presented for the Proceedings of the Symposium on Gravitation and BEC's Phenomenology from the Fourth Mexican Meeting on Mathematical and Experimental Physic

Perturbative analysis of gauged matrix models

We analyze perturbative aspects of gauged matrix models, including those where classically the gauge symmetry is partially broken. Ghost fields play a crucial role in the Feynman rules for these vacua. We use this formalism to elucidate the fact that nonperturbative aspects of [script N] = 1 gauge theories can be computed systematically using perturbative techniques of matrix models, even if we do not possess an exact solution for the matrix model. As examples we show how the Seiberg-Witten solution for [script N] = 2 gauge theory, the Montonen-Olive modular invariance for [script N] = 1*, and the superpotential for the Leigh-Strassler deformation of [script N] = 4 can be systematically computed in perturbation theory of the matrix model or gauge theory (even though in some of these cases an exact answer can also be obtained by summing up planar diagrams of matrix models)

Physics-inspired Performace Evaluation of a Structured Peer-to-Peer Overlay Network

In the majority of structured peer-to-peer overlay networks a graph with a desirable topology is constructed. In most cases, the graph is maintained by a periodic activity performed by each node in the graph to preserve the desirable structure in face of the continuous change of the set of nodes. The interaction of the autonomous periodic activities of the nodes renders the performance analysis of such systems complex and simulation of scales of interest can be prohibitive. Physicists, however, are accustomed to dealing with scale by characterizing a system using intensive variables, i.e. variables that are size independent. The approach has proved its usefulness when applied to satisfiability theory. This work is the first attempt to apply it in the area of distributed systems. The contribution of this paper is two-fold. First, we describe a methodology to be used for analyzing the performance of large scale distributed systems. Second, we show how we applied the methodology to find an intensive variable that describe the characteristic behavior of the Chord overlay network, namely, the ratio of the magnitude of perturbation of the network (joins/failures) to the magnitude of periodic stabilization of the network