32,850 research outputs found

    MAS: A versatile Landau-fluid eigenvalue code for plasma stability analysis in general geometry

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    We have developed a new global eigenvalue code, Multiscale Analysis for plasma Stabilities (MAS), for studying plasma problems with wave toroidal mode number n and frequency omega in a broad range of interest in general tokamak geometry, based on a five-field Landau-fluid description of thermal plasmas. Beyond keeping the necessary plasma fluid response, we further retain the important kinetic effects including diamagnetic drift, ion finite Larmor radius, finite parallel electric field, ion and electron Landau resonances in a self-consistent and non-perturbative manner without sacrificing the attractive efficiency in computation. The physical capabilities of the code are evaluated and examined in the aspects of both theory and simulation. In theory, the comprehensive Landau-fluid model implemented in MAS can be reduced to the well-known ideal MHD model, electrostatic ion-fluid model, and drift-kinetic model in various limits, which clearly delineates the physics validity regime. In simulation, MAS has been well benchmarked with theory and other gyrokinetic and kinetic-MHD hybrid codes in a manner of adopting the unified physical and numerical framework, which covers the kinetic Alfven wave, ion sound wave, low-n kink, high-n ion temperature gradient mode and kinetic ballooning mode. Moreover, MAS is successfully applied to model the Alfven eigenmode (AE) activities in DIII-D discharge #159243, which faithfully captures the frequency sweeping of RSAE, the tunneling damping of TAE, as well as the polarization characteristics of KBAE and BAAE being consistent with former gyrokinetic theory and simulation. With respect to the key progress contributed to the community, MAS has the advantage of combining rich physics ingredients, realistic global geometry and high computation efficiency together for plasma stability analysis in linear regime.Comment: 40 pages, 21 figure

    Quantum Mechanics Lecture Notes. Selected Chapters

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    These are extended lecture notes of the quantum mechanics course which I am teaching in the Weizmann Institute of Science graduate physics program. They cover the topics listed below. The first four chapter are posted here. Their content is detailed on the next page. The other chapters are planned to be added in the coming months. 1. Motion in External Electromagnetic Field. Gauge Fields in Quantum Mechanics. 2. Quantum Mechanics of Electromagnetic Field 3. Photon-Matter Interactions 4. Quantization of the Schr\"odinger Field (The Second Quantization) 5. Open Systems. Density Matrix 6. Adiabatic Theory. The Berry Phase. The Born-Oppenheimer Approximation 7. Mean Field Approaches for Many Body Systems -- Fermions and Boson

    Graph Convex Hull Bounds as generalized Jensen Inequalities

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    Jensen's inequality is ubiquitous in measure and probability theory, statistics, machine learning, information theory and many other areas of mathematics and data science. It states that, for any convex function f ⁣:KRf\colon K \to \mathbb{R} on a convex domain KRdK \subseteq \mathbb{R}^{d} and any random variable XX taking values in KK, E[f(X)]f(E[X])\mathbb{E}[f(X)] \geq f(\mathbb{E}[X]). In this paper, sharp upper and lower bounds on E[f(X)]\mathbb{E}[f(X)], termed "graph convex hull bounds", are derived for arbitrary functions ff on arbitrary domains KK, thereby strongly generalizing Jensen's inequality. Establishing these bounds requires the investigation of the convex hull of the graph of ff, which can be difficult for complicated ff. On the other hand, once these inequalities are established, they hold, just like Jensen's inequality, for any random variable XX. Hence, these bounds are of particular interest in cases where ff is fairly simple and XX is complicated or unknown. Both finite- and infinite-dimensional domains and codomains of ff are covered, as well as analogous bounds for conditional expectations and Markov operators.Comment: 12 pages, 1 figur

    Likelihood Asymptotics in Nonregular Settings: A Review with Emphasis on the Likelihood Ratio

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    This paper reviews the most common situations where one or more regularity conditions which underlie classical likelihood-based parametric inference fail. We identify three main classes of problems: boundary problems, indeterminate parameter problems -- which include non-identifiable parameters and singular information matrices -- and change-point problems. The review focuses on the large-sample properties of the likelihood ratio statistic. We emphasize analytical solutions and acknowledge software implementations where available. We furthermore give summary insight about the possible tools to derivate the key results. Other approaches to hypothesis testing and connections to estimation are listed in the annotated bibliography of the Supplementary Material

    Interpolating Discriminant Functions in High-Dimensional Gaussian Latent Mixtures

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    This paper considers binary classification of high-dimensional features under a postulated model with a low-dimensional latent Gaussian mixture structure and non-vanishing noise. A generalized least squares estimator is used to estimate the direction of the optimal separating hyperplane. The estimated hyperplane is shown to interpolate on the training data. While the direction vector can be consistently estimated as could be expected from recent results in linear regression, a naive plug-in estimate fails to consistently estimate the intercept. A simple correction, that requires an independent hold-out sample, renders the procedure minimax optimal in many scenarios. The interpolation property of the latter procedure can be retained, but surprisingly depends on the way the labels are encoded

    Path integrals and stochastic calculus

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    Path integrals are a ubiquitous tool in theoretical physics. However, their use is sometimes hindered by the lack of control on various manipulations -- such as performing a change of the integration path -- one would like to carry out in the light-hearted fashion that physicists enjoy. Similar issues arise in the field of stochastic calculus, which we review to prepare the ground for a proper construction of path integrals. At the level of path integration, and in arbitrary space dimension, we not only report on existing Riemannian geometry-based approaches that render path integrals amenable to the standard rules of calculus, but also bring forth new routes, based on a fully time-discretized approach, that achieve the same goal. We illustrate these various definitions of path integration on simple examples such as the diffusion of a particle on a sphere.Comment: 96 pages, 4 figures. New title, expanded introduction and additional references. Version accepted in Advandes in Physic

    Nonlocal error bounds for piecewise affine functions

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    The paper is devoted to a detailed analysis of nonlocal error bounds for nonconvex piecewise affine functions. We both improve some existing results on error bounds for such functions and present completely new necessary and/or sufficient conditions for a piecewise affine function to have an error bound on various types of bounded and unbounded sets. In particular, we show that any piecewise affine function has an error bound on an arbitrary bounded set and provide several types of easily verifiable sufficient conditions for such functions to have an error bound on unbounded sets. We also present general necessary and sufficient conditions for a piecewise affine function to have an error bound on a finite union of polyhedral sets (in particular, to have a global error bound), whose derivation reveals a structure of sublevel sets and recession functions of piecewise affine functions

    Worst-Case Control and Learning Using Partial Observations Over an Infinite Time-Horizon

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    Safety-critical cyber-physical systems require control strategies whose worst-case performance is robust against adversarial disturbances and modeling uncertainties. In this paper, we present a framework for approximate control and learning in partially observed systems to minimize the worst-case discounted cost over an infinite time-horizon. We model disturbances to the system as finite-valued uncertain variables with unknown probability distributions. For problems with known system dynamics, we construct a dynamic programming (DP) decomposition to compute the optimal control strategy. Our first contribution is to define information states that improve the computational tractability of this DP without loss of optimality. Then, we describe a simplification for a class of problems where the incurred cost is observable at each time-instance. Our second contribution is a definition of approximate information states that can be constructed or learned directly from observed data for problems with observable costs. We derive bounds on the performance loss of the resulting approximate control strategy

    Random Young towers and quenched decay of correlations for predominantly expanding multimodal circle maps

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    In this paper, we study the random dynamical system fωnf_\omega^n generated by a family of maps {fω0:S1S1}ω[ε,ε],\{f_{\omega_0}: \mathbb S^1 \to \mathbb S^1\}_{\omega \in [-\varepsilon,\varepsilon]}, $f_{\omega_0}(x) = \alpha \xi (x+\omega_0) +a\ (\mathrm{mod }\ 1),where where \xi: \mathbb S^1 \to \mathbb Risanondegeneratedmap, is a non-degenerated map, a\in \mathbb S^1,, \alpha,\varepsilon>0.Fixingaconstant. Fixing a constant c\in (0,1),weshowthatfor, we show that for \alphasufficientlylargeand sufficiently large and \varepsilon > \alpha^{-1+c},therandomdynamicalsystem the random dynamical system f_\omega^n$ presents a random Young tower structure and quenched decay of correlations.Comment: 38 pages, 0 figure
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