17,903 research outputs found

    Quantum Mechanics Lecture Notes. Selected Chapters

    Full text link
    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

    Fair Grading Algorithms for Randomized Exams

    Full text link
    This paper studies grading algorithms for randomized exams. In a randomized exam, each student is asked a small number of random questions from a large question bank. The predominant grading rule is simple averaging, i.e., calculating grades by averaging scores on the questions each student is asked, which is fair ex-ante, over the randomized questions, but not fair ex-post, on the realized questions. The fair grading problem is to estimate the average grade of each student on the full question bank. The maximum-likelihood estimator for the Bradley-Terry-Luce model on the bipartite student-question graph is shown to be consistent with high probability when the number of questions asked to each student is at least the cubed-logarithm of the number of students. In an empirical study on exam data and in simulations, our algorithm based on the maximum-likelihood estimator significantly outperforms simple averaging in prediction accuracy and ex-post fairness even with a small class and exam size

    Geometry of Rounding: Near Optimal Bounds and a New Neighborhood Sperner's Lemma

    Full text link
    A partition P\mathcal{P} of Rd\mathbb{R}^d is called a (k,ε)(k,\varepsilon)-secluded partition if, for every pRd\vec{p} \in \mathbb{R}^d, the ball B(ε,p)\overline{B}_{\infty}(\varepsilon, \vec{p}) intersects at most kk members of P\mathcal{P}. A goal in designing such secluded partitions is to minimize kk while making ε\varepsilon as large as possible. This partition problem has connections to a diverse range of topics, including deterministic rounding schemes, pseudodeterminism, replicability, as well as Sperner/KKM-type results. In this work, we establish near-optimal relationships between kk and ε\varepsilon. We show that, for any bounded measure partitions and for any d1d\geq 1, it must be that k(1+2ε)dk\geq(1+2\varepsilon)^d. Thus, when k=k(d)k=k(d) is restricted to poly(d){\rm poly}(d), it follows that ε=ε(d)O(lndd)\varepsilon=\varepsilon(d)\in O\left(\frac{\ln d}{d}\right). This bound is tight up to log factors, as it is known that there exist secluded partitions with k(d)=d+1k(d)=d+1 and ε(d)=12d\varepsilon(d)=\frac{1}{2d}. We also provide new constructions of secluded partitions that work for a broad spectrum of k(d)k(d) and ε(d)\varepsilon(d) parameters. Specifically, we prove that, for any f:NNf:\mathbb{N}\rightarrow\mathbb{N}, there is a secluded partition with k(d)=(f(d)+1)df(d)k(d)=(f(d)+1)^{\lceil\frac{d}{f(d)}\rceil} and ε(d)=12f(d)\varepsilon(d)=\frac{1}{2f(d)}. These new partitions are optimal up to O(logd)O(\log d) factors for various choices of k(d)k(d) and ε(d)\varepsilon(d). Based on the lower bound result, we establish a new neighborhood version of Sperner's lemma over hypercubes, which is of independent interest. In addition, we prove a no-free-lunch theorem about the limitations of rounding schemes in the context of pseudodeterministic/replicable algorithms

    PrivLava: Synthesizing Relational Data with Foreign Keys under Differential Privacy

    Full text link
    Answering database queries while preserving privacy is an important problem that has attracted considerable research attention in recent years. A canonical approach to this problem is to use synthetic data. That is, we replace the input database R with a synthetic database R* that preserves the characteristics of R, and use R* to answer queries. Existing solutions for relational data synthesis, however, either fail to provide strong privacy protection, or assume that R contains a single relation. In addition, it is challenging to extend the existing single-relation solutions to the case of multiple relations, because they are unable to model the complex correlations induced by the foreign keys. Therefore, multi-relational data synthesis with strong privacy guarantees is an open problem. In this paper, we address the above open problem by proposing PrivLava, the first solution for synthesizing relational data with foreign keys under differential privacy, a rigorous privacy framework widely adopted in both academia and industry. The key idea of PrivLava is to model the data distribution in R using graphical models, with latent variables included to capture the inter-relational correlations caused by foreign keys. We show that PrivLava supports arbitrary foreign key references that form a directed acyclic graph, and is able to tackle the common case when R contains a mixture of public and private relations. Extensive experiments on census data sets and the TPC-H benchmark demonstrate that PrivLava significantly outperforms its competitors in terms of the accuracy of aggregate queries processed on the synthetic data.Comment: This is an extended version of a SIGMOD 2023 pape

    Quantifying and Explaining Machine Learning Uncertainty in Predictive Process Monitoring: An Operations Research Perspective

    Full text link
    This paper introduces a comprehensive, multi-stage machine learning methodology that effectively integrates information systems and artificial intelligence to enhance decision-making processes within the domain of operations research. The proposed framework adeptly addresses common limitations of existing solutions, such as the neglect of data-driven estimation for vital production parameters, exclusive generation of point forecasts without considering model uncertainty, and lacking explanations regarding the sources of such uncertainty. Our approach employs Quantile Regression Forests for generating interval predictions, alongside both local and global variants of SHapley Additive Explanations for the examined predictive process monitoring problem. The practical applicability of the proposed methodology is substantiated through a real-world production planning case study, emphasizing the potential of prescriptive analytics in refining decision-making procedures. This paper accentuates the imperative of addressing these challenges to fully harness the extensive and rich data resources accessible for well-informed decision-making

    Tonelli Approach to Lebesgue Integration

    Full text link
    Leonida Tonelli devised an interesting and efficient method to introduce the Lebesgue integral. The details of this method can only be found in the original Tonelli paper and in an old italian course and solely for the case of the functions of one variable. We believe that it is woth knowing this method and here we present a complete account for functions of every number of variables

    Construction of radon chamber to expose active and passive detectors

    Get PDF
    In this research and development, we present the design and manufacture of a radon chamber (PUCP radon chamber), a necessary tool for the calibration of passive detectors, verification of the operation of active radon monitors as well as diffusion chamber calibration used in radon measurements in air, and soils. The first chapter is an introduction to describe radon gas and national levels of radon concentration given by many organizations. Parameters that influence the calibration factor of the LR 115 type 2 film detector are studied, such as the energy window, critical angle, and effective volumes. Those are strongly related to the etching processes and counting of tracks all seen from a semi-empirical approach studied in the second chapter. The third chapter presents a review of some radon chambers that have been reported in the literature, based on their size and mode of operation as well as the radon source they use. The design and construction of the radon chamber are presented, use of uranium ore (autunite) as a chamber source is also discussed. In chapter fourth, radon chamber characterization is presented through leakage lambda, homogeneity of radon concentration, regimes-operation modes, and the saturation concentrations that can be reached. Procedures and methodology used in this work are contained in the fifth chapter and also some uses and applications of the PUCP radon chamber are presented; the calibration of cylindrical metallic diffusion chamber based on CR-39 chips detectors taking into account overlapping effect; transmission factors of gaps and pinhole for the same diffusion chambers are determined; permeability of glass fiber filter for 222Rn is obtained after reach equilibrium through Ramachandran model and taking into account a partition function as the rate of track density. The results of this research have been published in indexed journals. Finally, the conclusion and recommendations that reflect the fulfillment aims of this thesis are presented

    A study of uncertainty quantification in overparametrized high-dimensional models

    Full text link
    Uncertainty quantification is a central challenge in reliable and trustworthy machine learning. Naive measures such as last-layer scores are well-known to yield overconfident estimates in the context of overparametrized neural networks. Several methods, ranging from temperature scaling to different Bayesian treatments of neural networks, have been proposed to mitigate overconfidence, most often supported by the numerical observation that they yield better calibrated uncertainty measures. In this work, we provide a sharp comparison between popular uncertainty measures for binary classification in a mathematically tractable model for overparametrized neural networks: the random features model. We discuss a trade-off between classification accuracy and calibration, unveiling a double descent like behavior in the calibration curve of optimally regularized estimators as a function of overparametrization. This is in contrast with the empirical Bayes method, which we show to be well calibrated in our setting despite the higher generalization error and overparametrization
    corecore