Real analysis

In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability. Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions. Scope Construction of the real numbers Construction of the real numbers The theorems of real analysis rely on the properties of the real number system, which must be established. The real number system consists of an uncountable set (\mathbb{R}), together with two binary operations denoted + and ⋅, and an order denoted

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Fast deterministic and randomized algorithms for low-rank approximation, matrix functions, and trace estimation

Alice Cortinovis

In this thesis we propose and analyze algorithms for some numerical linear algebra tasks: finding low-rank approximations of matrices, computing matrix functions, and estimating the trace of matrices.In the first part, we consider algorithms for building low-rank approximations of a matrix from some rows or columns of the matrix itself. We prove a priori error bounds for a greedy algorithm for cross approximation and we develop a faster and more efficient variant of an existing algorithm for column subset selection. Moreover, we present a new deterministic polynomial-time algorithm that gives a cross approximation which is quasi-optimal in the Frobenius norm. The second part of the thesis is concerned with matrix functions. We develop a divide-and-conquer algorithm for computing functions of matrices that are banded, hierarchically semiseparable, or have some other off-diagonal low-rank structure. An important building block of our approach is an existing algorithm for updating the function of a matrix that undergoes a low-rank modification (update), for which we present new convergence results. The convergence analysis of our divide-and-conquer algorithm is related to polynomial or rational approximation of the function.In the third part we consider the problem of approximating the trace of a matrix which is available indirectly, through matrix-vector multiplications. We analyze a stochastic algorithm, the Hutchinson trace estimator, for which we prove tail bounds for symmetric (indefinite) matrices. Then we apply our results to the computation of the (log)determinants of symmetric positive definite matrices.

EPFL2022

Strictly Real Fundamental Theorem Of Algebra Using Polynomial Interlacing

Soham Basu

Without resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra. The proof uses interlacing of bivariate polynomials similar to Gauss's first proof of the fundamental theorem of algebra using complex numbers, but in a different context of division residues of strictly real polynomials. This shows the sufficiency of basic real analysis as the minimal platform to prove the fundamental theorem of algebra.

CAMBRIDGE UNIV PRESS2021

A Speech-based Productive Engagement Metric for Real-time Human-Robot Interaction in Collaborative Educational Contexts

Barbara Bruno, Pierre Dillenbourg, Jauwairia Nasir

One of the major challenges in educational contexts is to correctly interpret the student's learning process in real-time, which is a necessary pre-requisite for timely and appropriate interventions. In educational HRI, multiple solutions have been proposed to endow robots with this key ability, typically relying on observable proxies such as in-task performance, affective behavior, engagement with the robot, etc. In this methodology paper, we propose and validate a metric for the real-time analysis of the behaviour of learners, allowing to assess whether they are engaged in meaningful learning behaviours. Specifically, building on the previously proposed concept of Productive Engagement, that inherently links learning with engagement, we hereby propose methods to quantify and compute it reliably in real-time. The training and testing of the methods is done using the open access PE-HRI-temporal dataset, that provides team level multi-modal temporal behavioral data, built from a study done with 68 students (34 teams) in a collaborative human-human educational activity mediated by a robot.

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