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Register a new userOperational approach to the topological structure of the physical space
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Abstract axiomatic formulation of mathematical structures are extensively used to describe our physical world. We take here the reverse way. By making basic assumptions as starting point, we reconstruct some features of both geometry and topology in a fully operational manner. Curiously enough, primitive concepts such as points, spaces, straight lines, planes are all defined within our formalism. Our construction breaks down with the usual literature, as our axioms have deep connection with nature. Besides that, we hope this operational approach could also be of pedagogical interest.
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This article is an updated version of an entry of the Encyclopedia of Mathematics Education (2018). In the same time, it is the seed of the HAL collection DAD-MULTILINGUAL, constituted by the translation of this entry in various languages.The documentational approach to didactics is a theory in mathematics education. Its first aim is to understand teachers professional development by studying their interactions with the resources they use and design in/for their teaching. In this text we briefly describe the emergence of the approach, its theoretical sources, its main concepts and the associated methodology. We illustrate these aspects with examples from different research projects. This synthetic presentation is written for researchers, but also for non-specialists (e.g. master students) interested in a first discovery of the documentational approach.
Pulli kolam is a ubiquitous art form in south India. It involves drawing a line looped around a collection of dots (pullis) place on a plane such that three mandatory rules are followed: all line orbits should be closed, all dots are encircled and no two lines can overlap over a finite length. The mathematical foundation for this art form has attracted attention over the years. In this work, we propose a simple 5-step topological method by which one can systematically draw all possible kolams for any number of dots N arranged in any spatial configuration on a surface.
Given two random variables $X$ and $Y$, an operational approach is undertaken to quantify the ``leakage of information from $X$ to $Y$. The resulting measure $mathcal{L}(X !! to !! Y)$ is called emph{maximal leakage}, and is defined as the multiplicative increase, upon observing $Y$, of the probability of correctly guessing a randomized function of $X$, maximized over all such randomized functions. A closed-form expression for $mathcal{L}(X !! to !! Y)$ is given for discrete $X$ and $Y$, and it is subsequently generalized to handle a large class of random variables. The resulting properties are shown to be consistent with an axiomatic view of a leakage measure, and the definition is shown to be robust to variations in the setup. Moreover, a variant of the Shannon cipher system is studied, in which performance of an encryption scheme is measured using maximal leakage. A single-letter characterization of the optimal limit of (normalized) maximal leakage is derived and asymptotically-optimal encryption schemes are demonstrated. Furthermore, the sample complexity of estimating maximal leakage from data is characterized up to subpolynomial factors. Finally, the emph{guessing} framework used to define maximal leakage is used to give operational interpretations of commonly used leakage measures, such as Shannon capacity, maximal correlation, and local differential privacy.
The notions of error and disturbance appearing in quantum uncertainty relations are often quantified by the discrepancy of a physical quantity from its ideal value. However, these real and ideal values are not the outcomes of simultaneous measurements, and comparing the values of unmeasured observables is not necessarily meaningful according to quantum theory. To overcome these conceptual difficulties, we take a different approach and define error and disturbance in an operational manner. In particular, we formulate both in terms of the probability that one can successfully distinguish the actual measurement device from the relevant hypothetical ideal by any experimental test whatsoever. This definition itself does not rely on the formalism of quantum theory, avoiding many of the conceptual difficulties of usual definitions. We then derive new Heisenberg-type uncertainty relations for both joint measurability and the error-disturbance tradeoff for arbitrary observables of finite-dimensional systems, as well as for the case of position and momentum. Our relations may be directly applied in information processing settings, for example to infer that devices which can faithfully transmit information regarding one observable do not leak any information about conjugate observables to the environment. We also show that Englerts wave-particle duality relation [PRL 77, 2154 (1996)] can be viewed as an error-disturbance uncertainty relation.
The data from ground based gravitational-wave detectors such as Advanced LIGO and Virgo must be calibrated to convert the digital output of photodetectors into a relative displacement of the test masses in the detectors, producing the quantity of interest for inference of astrophysical gravitational wave sources. Both statistical uncertainties and systematic errors are associated with the calibration process, which would in turn affect the analysis of detected sources, if not accounted for. Currently, source characterization algorithms either entirely neglect the possibility of calibration uncertainties or account for them in a way that does not use knowledge of the calibration process itself. We present physiCal, a new approach to account for calibration errors during the source characterization step, which directly uses all the information available about the instrument calibration process. Rather than modeling the overall detectors response function, we consider the individual components that contribute to the response. We implement this method and apply it to the compact binaries detected by LIGO and Virgo during the second observation run, as well as to simulated binary neutron stars for which the sky position and distance are known exactly. We find that the physiCal model performs as well as the method currently used within the LIGO-Virgo collaboration, but additionally it enables improving the measurement of specific components of the instrument control through astrophysical calibration.
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